sympy/source/doc/src/modules/solvers/solveset.rst

.. _solveset:

Solveset
========

.. module:: sympy.solvers.solveset

This is the official documentation of the ``solveset`` module in solvers.

.. note::

   For a beginner-friendly guide focused on solving common types of equations,
   refer to :ref:`solving-guide`.

How solveset() is different from solve()
----------------------------------------

SymPy already has a pretty powerful ``solve`` function. But it has different
goals. For example:

* It has a generalized output interface to handle various types of solutions.
  It needs to return many types of solutions consistently. It can deal with
  univariate as well as multivariate equations as well as other kinds
  of variability due to the system of equations and options passed into it,
  producing associated outputs. See :ref:`solve_output` for more details.

* The input API has a lot of parameters. It is a facade around
  more specialized solvers. *This feature of* ``solve`` *could be viewed as analogous
  to how* ``simplify`` *is a facade around more specialized simplification functions.*

  See :func:`~sympy.solvers.solvers.solve` for more details about the available parameters
  and output options.

* There are cases where ``solve`` returns an empty list.

  This might mean that there are no solutions or no solution could be found
  given its currently supported features. In other cases, there is no way
  (given the output interface of ``solve``) to communicate whether or not:

  * all possible solutions to the system were found
  * or that there are provably no solutions
  * or that the real set of solutions are finite or infinite
  * etc.

* ``solve`` may return a few solutions when more solutions (potentially
  infinitely many) also exist.

  It is to some extent inherent to the API of ``solve`` that it does not
  have a way to represent these different cases distinctly.

Whereas ``solveset`` returns a ``Set`` representing all of the solutions
of a univariate equation.


.. _why-solveset:

Why Solveset?
-------------

``solveset`` has a consistent input and output interface:

* ``solveset`` returns a ``Set`` object in a way that takes care of all types of output.

  For cases where it does not "know" all of the solutions a ``ConditionSet``
  with a partial solution is returned.

  For input it only takes the equation, the variables to solve for and the optional
  argument ``domain`` over which the equation is to be solved.

  See :ref:`solveset-input-api` for more details about the available
  parameters and :ref:`why-set-output` below for more about ``Set`` output.

* ``solveset`` can return infinitely many solutions. For example solving for
  `\sin{(x)} = 0` returns `\{2 n \pi | n \in \mathbb{Z}\} \cup \{2 n \pi + \pi | n \in \mathbb{Z}\}`,
  whereas ``solve`` only returns `[0, \pi]`.

* ``solveset`` has a clear code level and interface level separation between solvers
  for equations in the complex and real domains.

  For example solving `e^x = 1` when `x` is to be solved in the complex domain, returns
  the set of all solutions, that is `\{2 n i \pi | n \in \mathbb{Z}\}`, whereas
  if `x` is to be solved in the real domain then only `\{0\}` is returned.

* As mentioned above, ``solveset`` will return a ``ConditionSet`` where not all
  parts of an equation can be solved. Consider this example.

  ``expr = (x**2 - 1) * (x**5 + y*x + 1)`` where ``x`` and ``y`` are defined as being
  in `\mathbb{C}`.

  ``solve(expr, [x], dict=True)`` gives us the solutions it can find. However it gives
  us no indication that there may be more solutions that it did not return.

  ``[{x: -1},{x: 1}]``

  However, with ``solveset(expr, x)`` the partial solution becomes clear. The quadratic factor
  was solved, but the quintic factor is not solvable and so a ``ConditionSet`` is returned.

  `\displaystyle \left\{-1, 1\right\} \cup \left\{x\; \middle|\; x \in \mathbb{C} \wedge x^{5} + x y + 1 = 0 \right\}`


.. _why-set-output:

Why do we use Sets as an output type?
-------------------------------------

SymPy has a well developed sets module, which can represent most of the set
containers in mathematics such as:

 * :class:`~.FiniteSet`

   Represents a finite set of discrete numbers.


 * :class:`~.Interval`

   Represents a real interval as a set.


 * :class:`~.ProductSet`

   Represents a Cartesian product of sets.


 * :class:`~.ImageSet`

   Represents the image of a set under a mathematical function

    >>> from sympy import ImageSet, S, Lambda
    >>> from sympy.abc import x
    >>> squares = ImageSet(Lambda(x, x**2), S.Naturals)  # {x**2 for x in N}
    >>> 4 in squares
    True

 * :class:`~.ComplexRegion`

   Represents the set of all complex numbers in a region in the Argand plane.


 * :class:`~.ConditionSet`

   Represents the set of elements, which satisfies a given condition.


Also, the predefined set classes such as:

 * :class:`~.Naturals`, $\mathbb{N}$

   Represents the natural numbers (or counting numbers), which are all
   positive integers starting from 1.


 * :class:`~.Naturals0`, $\mathbb{N_0}$

   Represents the whole numbers, which are all the non-negative integers,
   inclusive of 0.


 * :class:`~.Integers`, $\mathbb{Z}$

   Represents all integers: positive, negative and zero.


 * :class:`~.Reals`, $\mathbb{R}$

   Represents the set of all real numbers.


 * :class:`~.Complexes`, $\mathbb{C}$

   Represents the set of all complex numbers.


 * :class:`~.EmptySet`, $\emptyset$

   Represents the empty set.

 The above six sets are available as Singletons, like ``S.Integers``.


It is capable of most of the set operations in mathematics:

 * ``Union``
 * ``Intersection``
 * ``Complement``
 * ``SymmetricDifference``

The main reason for using sets as output to solvers is that it can consistently
represent many types of solutions. For the single variable case it can represent:

 * No solution (by the empty set).

 * Finitely many solutions (by ``FiniteSet``).

 * Infinitely many solutions, both countably and uncountably infinite solutions
   (using the ``ImageSet`` module).

 * ``Interval``

 * There can also be bizarre solutions to equations like the set of rational
   numbers.

No other Python object (list, dictionary, generator, Python sets) provides the
flexibility of mathematical sets which our sets module tries to emulate. The
second reason to use sets is that they are close to the entities which
mathematicians deal with and it makes it easier to reason about them.  Set
objects conform to Pythonic conventions when possible, i.e., ``x in A`` and
``for i in A`` both work when they can be computed. Another advantage of using
objects closer to mathematical entities is that the user won't have to "learn"
our representation and she can have her expectations transferred from her
mathematical experience.

For the multivariate case we represent solutions as a set of points in a
n-dimensional space and a point is represented by a ``FiniteSet`` of ordered
tuples, which is a point in `\mathbb{R}^n` or `\mathbb{C}^n`.

Please note that, the general ``FiniteSet`` is unordered, but a ``FiniteSet``
with a tuple as its only argument becomes ordered, since a tuple is ordered.
So the order in the tuple is mapped to a pre-defined order of variables
while returning solutions.

For example:

 >>> from sympy import FiniteSet
 >>> FiniteSet(1, 2, 3)   # Unordered
 {1, 2, 3}
 >>> FiniteSet((1, 2, 3))  # Ordered
 {(1, 2, 3)}


Why not use dicts as output?

  Dictionary are easy to deal with programmatically but mathematically they are
  not very precise and use of them can quickly lead to inconsistency and a lot
  of confusion. For example:

  * There are a lot of cases where we don't know the complete solution and we
    may like to output a partial solution, consider the equation `fg = 0`. The
    solution of this equation is the union of the solution of the following
    two equations: `f = 0`, `g = 0`. Let's say that we are able to solve
    `f = 0` but solving `g = 0` isn't supported yet. In this case we cannot
    represent partial solution of the given equation `fg = 0` using dicts.
    This problem is solved with sets using a ``ConditionSet`` object:

    `sol_f \cup \{x | x ∊ \mathbb{R} ∧ g = 0\}`, where `sol_f` is the solution
    of the equation `f = 0`.

  * Using a dict may lead to surprising results like:

    - ``solve(Eq(x**2, 1), x) != solve(Eq(y**2, 1), y)``


      Mathematically, this doesn't make sense. Using ``FiniteSet`` here
      solves the problem.

  * It also cannot represent solutions for equations like `|x| < 1`, which is
    a disk of radius 1 in the Argand Plane. This problem is solved using
    complex sets implemented as ``ComplexRegion``.

.. _solveset-input-api:

Input API of ``solveset``
-------------------------

``solveset`` has simpler input API, unlike ``solve``. It takes a maximum of
three arguments:

``solveset(equation, variable=None, domain=S.Complexes)``

Equation
   The equation to solve.
Variable
   The variable for which the equation is to be solved.
Domain
   The domain in which the equation is to be solved.

``solveset`` removes the ``flags`` argument of ``solve``, which had made the
input API more complicated and output API inconsistent.

.. _solveset-domain-argument:

What is this domain argument about?
-----------------------------------

Solveset is designed to be independent of the assumptions on the variable being
solved for and instead, uses the ``domain`` argument to decide the solver to
dispatch the equation to, namely ``solveset_real`` or ``solveset_complex``.
It's unlike the old ``solve`` which considers the assumption on the variable.

    >>> from sympy import solveset, S
    >>> from sympy.abc import x
    >>> solveset(x**2 + 1, x) # domain=S.Complexes is default
    {-I, I}
    >>> solveset(x**2 + 1, x, domain=S.Reals)
    EmptySet

What are the general methods employed by solveset to solve an equation?
-----------------------------------------------------------------------

 Solveset uses various methods to solve an equation, here is a brief overview
 of the methodology:

 * The ``domain`` argument is first considered to know the domain in which
   the user is interested to get the solution.


 * If the given function is a relational (``>=``, ``<=``, ``>``, ``<``), and
   the domain is real, then ``solve_univariate_inequality`` and solutions are
   returned. Solving for complex solutions of inequalities, like `x^2 < 0`
   is not yet supported.


 * Based on the ``domain``, the equation is dispatched to one of the two
   functions ``solveset_real`` or ``solveset_complex``, which solves the
   given equation in the complex or real domain, respectively.


 * If the given expression is a product of two or more functions,
   like say `gh = 0`, then the solution to the given equation is the Union
   of the solution of the equations `g = 0` and `h = 0`, if and only if both
   `g` and `h` are finite for a finite input. So, the solution is built up
   recursively.


 * If the function is trigonometric or hyperbolic, the function
   ``_solve_real_trig`` is called, which solves it by converting it to complex
   exponential form.


 * The function is now checked if there is any instance of a ``Piecewise``
   expression, if it is, then it's converted to explicit expression and
   set pairs and then solved recursively.


 * The respective solver now tries to invert the equation using the routines
   ``invert_real`` and ``invert_complex``. These routines are based on the
   concept of mathematical inverse (though not exactly). It reduces the
   real/complex valued equation `f(x) = y` to a set of equations:
   `\{g(x)  = h_1(y), g(x) = h_2(y), ..., g(x) = h_n(y) \}` where `g(x)` is a
   simpler function than `f(x)`. There is some work needed to be done in
   this to find invert of more complex expressions.


 * After the invert, the equations are checked for radical or Abs (Modulus),
   then the method ``_solve_radical`` tries to simplify the radical, by
   removing it using techniques like squaring, cubing etc, and ``_solve_abs``
   solves nested Modulus by considering the positive and negative variants,
   iteratively.


 * If none of the above method is successful, then methods of polynomial is
   used as follows:

   - The method to solve the rational function, ``_solve_as_rational``, is
     called.  Based on the domain, the respective poly solver
     ``_solve_as_poly_real`` or ``_solve_as_poly_complex`` is called to solve
     ``f`` as a polynomial.

   - The underlying method ``_solve_as_poly`` solves the equation using
     polynomial techniques if it's already a polynomial equation or, with
     a change of variables, can be made so.


 * The final solution set returned by ``solveset`` is the intersection of the
   set of solutions found above and the input domain.

.. Remember to change the above part when the new solver is implemented.


How do we manipulate and return an infinite solution?
-----------------------------------------------------

 * In the real domain, we use our ``ImageSet`` class in the sets module to
   return infinite solutions. ``ImageSet`` is an image of a set under
   a mathematical function. For example, to represent the solution
   of the equation `\sin{(x)} = 0`, we can use the ``ImageSet`` as:


   >>> from sympy import ImageSet, Lambda, pi, S, Dummy, pprint
   >>> n = Dummy('n')
   >>> pprint(ImageSet(Lambda(n, 2*pi*n), S.Integers), use_unicode=True)
   {2⋅n⋅π │ n ∊ ℤ}


   Where ``n`` is a dummy variable. It is basically the image of the
   set of integers under the function `2\pi n`.

 * In the complex domain, we use complex sets, which are implemented as the
   ``ComplexRegion`` class in the sets module, to represent infinite
   solution in the Argand plane. For example to represent the solution
   of the equation `|z| = 1`, which is a unit circle, we can use
   the ``ComplexRegion`` as:


   >>> from sympy import ComplexRegion, FiniteSet, Interval, pi, pprint
   >>> pprint(ComplexRegion(FiniteSet(1)*Interval(0, 2*pi), polar=True), use_unicode=True)
   {r⋅(ⅈ⋅sin(θ) + cos(θ)) │ r, θ ∊ {1} × [0, 2⋅π)}


   Where the ``FiniteSet`` in the ``ProductSet`` is the range of the value
   of `r`, which is the radius of the circle and the ``Interval`` is the range
   of `\theta`, the angle from the `x` axis representing a unit circle in the
   Argand plane.

   Note: We also have non-polar form notation for representing solution
   in rectangular form. For example, to represent first two quadrants in the
   Argand plane, we can write the ``ComplexRegion`` as:


   >>> from sympy import ComplexRegion, Interval, pi, oo, pprint
   >>> pprint(ComplexRegion(Interval(-oo, oo)*Interval(0, oo)), use_unicode=True)
   {x + y⋅ⅈ │ x, y ∊ (-∞, ∞) × [0, ∞)}


   where the Intervals are the range of `x` and `y` for the set of complex
   numbers `x + iy`.


How does ``solveset`` ensure that it is not returning any wrong solution?
--------------------------------------------------------------------------

 Solvers in a Computer Algebra System are based on heuristic algorithms,
 so it's usually very hard to ensure 100% percent correctness, in every
 possible case. However there are still a lot of cases where we can ensure
 correctness. Solveset tries to verify correctness wherever it can. For
 example:

 Consider the equation `|x| = n`. A naive method to solve this equation
 would return ``{-n, n}`` as its solution, which is not correct since
 ``{-n, n}`` can be its solution if and only if ``n`` is positive.
 Solveset returns this information as well to ensure correctness.

    >>> from sympy import symbols, S, pprint, solveset
    >>> x, n = symbols('x, n')
    >>> pprint(solveset(abs(x) - n, x, domain=S.Reals), use_unicode=True)
    {x │ x ∊ {-n, n} ∧ (n ∈ [0, ∞))}

 Though, there still a lot of work needs to be done in this regard.


Search based solver and step-by-step solution
---------------------------------------------

 Note: This is under Development.

 After the introduction of :py:class:`~sympy.sets.conditionset.ConditionSet`, the
 solving of equations can be seen as set transformations. Here is an abstract
 view of the things we can do to solve equations.

 * Apply various set transformations on the given set.
 * Define a metric of the usability of solutions, or a notion of some
   solutions being better than others.
 * Different transformations would be the nodes of a tree.
 * Suitable searching techniques could be applied to get the best solution.

 ``ConditionSet`` gives us the ability to represent unevaluated equations and
 inequalities in forms like `\{x|f(x)=0; x \in S\}` and `\{x|f(x)>0; x \in S\}`
 but a more powerful thing about ``ConditionSet`` is that it allows us to write
 the intermediate steps as set to set transformation. Some of the transformations
 are:

 * Composition: `\{x|f(g(x))=0;x \in S\} \Rightarrow \{x|g(x)=y; x \in S, y \in \{z|f(z)=0; z \in S\}\}`

 * Polynomial Solver: `\{x | P(x) = 0;x \in S\} \Rightarrow  \{x_1,x_2, ... ,x_n\} \cap S`,
                      where `x_i` are roots of `P(x)`.

 * Invert solver: `\{x|f(x)=0;x \in S\} \Rightarrow  \{g(0)| \text{ all g such that } f(g(x)) = x\}`

 * logcombine: `\{x| \log(f(x)) + \log(g(x));x \in S\}`
               `\Rightarrow  \{x| \log(f(x).g(x)); x \in S\} \text{ if } f(x) > 0 \text{ and } g(x) > 0`
               `\Rightarrow  \{x| \log(f(x)) + \log(g(x));x \in S\} \text{ otherwise}`

 * product solve: `\{x|f(x)g(x)=0; x \in S\}`
                  `\Rightarrow  \{x|f(x)=0; x \in S\} U \{x|g(x)=0; x \in S\}`
                  `\text{ given } f(x) \text{ and } g(x) \text{ are bounded.}`
                  `\Rightarrow  \{x|f(x)g(x)=0; x \in S\}, \text{ otherwise}`

 Since the output type is same as the input type any composition of these
 transformations is also a valid transformation. And our aim is to find
 the right sequence of compositions (given the atoms) which transforms
 the given condition set to a set which is not a condition set i.e.,
 FiniteSet, Interval, Set of Integers and their Union, Intersection,
 Complement or ImageSet. We can assign a cost function to each set,
 such that, the more desirable that form of set is to us, the less the value
 of the cost function. This way our problem is now reduced to finding the path
 from the initial ConditionSet to the lowest valued set on a graph where
 the atomic transformations forms the edges.


How do we deal with cases where only some of the solutions are known?
---------------------------------------------------------------------

 Creating a universal equation solver, which can solve each and every equation
 we encounter in mathematics is an ideal case for solvers in a Computer
 Algebra System. When cases which are not solved or can only be solved
 incompletely, a ``ConditionSet`` is used and acts as an unevaluated solveset
 object.

 Note that, mathematically, finding a complete set of solutions for an
 equation is undecidable. See `Richardson's theorem
 <https://en.wikipedia.org/wiki/Richardson%27s_theorem>`_.

 ``ConditionSet`` is basically a Set of elements which satisfy a given
 condition. For example, to represent the solutions of the equation in
 the real domain:

 .. math::  (x^2 - 4)(\sin(x) + x)

 We can represent it as:

 `\{-2, 2\} ∪ \{x | x \in \mathbb{R} ∧ x + \sin(x) = 0\}`


How are symbolic parameters handled in solveset?
------------------------------------------------

 Solveset is in its initial phase of development, so the symbolic parameters
 aren't handled well for all the cases, but some work has been done in this
 regard to depict our ideology towards symbolic parameters. As an example,
 consider the solving of `|x| = n` for real `x`, where `n` is a symbolic
 parameter. Solveset returns the value of `x` considering the domain of the
 symbolic parameter `n` as well:

 .. math:: ([0, \infty) \cap \{n\}) \cup ((-\infty, 0] \cap \{-n\}).

 This simply means `n` is the solution only when it belongs to the
 ``Interval`` `[0, \infty)` and `-n` is the solution only when `-n` belongs to
 the ``Interval`` `(- \infty, 0]`.

 There are other cases to address too, like solving `2^x + (a - 2)` for `x`
 where `a` is a symbolic parameter.  As of now, It returns the solution as an
 intersection with `\mathbb{R}`, which is trivial, as it doesn't reveal the
 domain of `a` in the solution.

 Recently, we have also implemented a function to find the domain of the
 expression in a FiniteSet (Intersection with the interval) in which it is
 not-empty. It is a useful addition for dealing with symbolic parameters.
 For example:

    >>> from sympy import Symbol, FiniteSet, Interval, not_empty_in, sqrt, oo
    >>> from sympy.abc import x
    >>> not_empty_in(FiniteSet(x/2).intersect(Interval(0, 1)), x)
    Interval(0, 2)
    >>> not_empty_in(FiniteSet(x, x**2).intersect(Interval(1, 2)), x)
    Union(Interval(1, 2), Interval(-sqrt(2), -1))


References
----------

 .. [1] https://github.com/sympy/sympy/wiki/GSoC-2015-Ideas/7abb76ffed50425299b9065129ae87261668a0f7#user-content-solvers
 .. [2] https://github.com/sympy/sympy/wiki/GSoC-2014-Application-Harsh-Gupta:-Solvers
 .. [3] https://github.com/sympy/sympy/wiki/GSoC-2015-Application-AMiT-Kumar--Solvers-:-Extending-Solveset
 .. [5] https://iamit.in/blog/
 .. [6] https://github.com/sympy/sympy/pull/2948 : Action Plan for improving solvers.
 .. [7] https://github.com/sympy/sympy/issues/6659 : ``solve()`` is a giant mess
 .. [8] https://github.com/sympy/sympy/pull/7523 : ``solveset`` PR
 .. [9] https://groups.google.com/forum/#!topic/sympy/-SIbX0AFL3Q
 .. [10] https://github.com/sympy/sympy/pull/9696
 .. [11] https://en.wikipedia.org/wiki/Richardson%27s_theorem


Solveset Module Reference
-------------------------

Use :func:`solveset` to solve equations or expressions (assumed to be equal to 0) for a single variable.
Solving an equation like `x^2 == 1` can be done as follows::

    >>> from sympy import solveset
    >>> from sympy import Symbol, Eq
    >>> x = Symbol('x')
    >>> solveset(Eq(x**2, 1), x)
    {-1, 1}

Or one may manually rewrite the equation as an expression equal to 0::

    >>> solveset(x**2 - 1, x)
    {-1, 1}

The first argument for :func:`solveset` is an expression (equal to zero) or an equation and the second argument
is the symbol that we want to solve the equation for.

.. autofunction:: sympy.solvers.solveset::solveset

.. autofunction:: sympy.solvers.solveset::solveset_real

.. autofunction:: sympy.solvers.solveset::solveset_complex

.. autofunction:: sympy.solvers.solveset::invert_real

.. autofunction:: sympy.solvers.solveset::invert_complex

.. autofunction:: sympy.solvers.solveset::domain_check

.. autofunction:: sympy.solvers.solveset::solvify

.. autofunction:: sympy.solvers.solveset::linear_eq_to_matrix

.. autofunction:: sympy.solvers.solveset::linsolve

.. autofunction:: sympy.solvers.solveset::nonlinsolve


transolve
^^^^^^^^^

.. autofunction:: sympy.solvers.solveset::_transolve

.. autofunction:: sympy.solvers.solveset::_is_exponential

.. autofunction:: sympy.solvers.solveset::_solve_exponential

.. autofunction:: sympy.solvers.solveset::_solve_logarithm

.. autofunction:: sympy.solvers.solveset::_is_logarithmic


Diophantine Equations (DEs)
---------------------------

See :ref:`diophantine-docs`

Inequalities
------------

See :ref:`inequality-docs`

Ordinary Differential equations (ODEs)
--------------------------------------

See :ref:`ode-docs`.

Partial Differential Equations (PDEs)
-------------------------------------

See :ref:`pde-docs`.