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|---|---|---|
169,709 | import sys
import os
from pathlib import Path
import io
def asstr(s):
if isinstance(s, bytes):
return s.decode('latin1')
return str(s) | null |
169,710 | import sys
import os
from pathlib import Path
import io
def isfileobj(f):
return isinstance(f, (io.FileIO, io.BufferedReader, io.BufferedWriter)) | null |
169,711 | import sys
import os
from pathlib import Path
import io
def open_latin1(filename, mode='r'):
return open(filename, mode=mode, encoding='iso-8859-1') | null |
169,712 | import sys
import os
from pathlib import Path
import io
def sixu(s):
return s | null |
169,713 | import sys
import os
from pathlib import Path
import io
import sys
if sys.version_info < (3, 8):
try:
import pickle5 as pickle # noqa: F401
from pickle5 import Pickler # noqa: F401
except ImportError:
import pickle # noqa: F401
# Use the Python pickler for old CPython versions
from pickle import _Pickler as Pickler # noqa: F401
else:
import pickle # noqa: F401
# Pickler will the C implementation in CPython and the Python
# implementation in PyPy
from pickle import Pickler # noqa: F401
def getexception():
return sys.exc_info()[1] | null |
169,714 | import sys
import os
from pathlib import Path
import io
unicode = str
def asbytes(s):
def asbytes_nested(x):
if hasattr(x, '__iter__') and not isinstance(x, (bytes, unicode)):
return [asbytes_nested(y) for y in x]
else:
return asbytes(x) | null |
169,715 | import sys
import os
from pathlib import Path
import io
unicode = str
def asunicode(s):
if isinstance(s, bytes):
return s.decode('latin1')
return str(s)
def asunicode_nested(x):
if hasattr(x, '__iter__') and not isinstance(x, (bytes, unicode)):
return [asunicode_nested(y) for y in x]
else:
return asunicode(x) | null |
169,716 | import sys
import os
from pathlib import Path
import io
class Path(PurePath):
def __new__(cls: Type[_P], *args: Union[str, _PathLike], **kwargs: Any) -> _P: ...
def __enter__(self: _P) -> _P: ...
def __exit__(
self, exc_type: Optional[Type[BaseException]], exc_value: Optional[BaseException], traceback: Optional[TracebackType]
) -> Optional[bool]: ...
def cwd(cls: Type[_P]) -> _P: ...
def stat(self) -> os.stat_result: ...
def chmod(self, mode: int) -> None: ...
def exists(self) -> bool: ...
def glob(self: _P, pattern: str) -> Generator[_P, None, None]: ...
def group(self) -> str: ...
def is_dir(self) -> bool: ...
def is_file(self) -> bool: ...
if sys.version_info >= (3, 7):
def is_mount(self) -> bool: ...
def is_symlink(self) -> bool: ...
def is_socket(self) -> bool: ...
def is_fifo(self) -> bool: ...
def is_block_device(self) -> bool: ...
def is_char_device(self) -> bool: ...
def iterdir(self: _P) -> Generator[_P, None, None]: ...
def lchmod(self, mode: int) -> None: ...
def lstat(self) -> os.stat_result: ...
def mkdir(self, mode: int = ..., parents: bool = ..., exist_ok: bool = ...) -> None: ...
# Adapted from builtins.open
# Text mode: always returns a TextIOWrapper
def open(
self,
mode: OpenTextMode = ...,
buffering: int = ...,
encoding: Optional[str] = ...,
errors: Optional[str] = ...,
newline: Optional[str] = ...,
) -> TextIOWrapper: ...
# Unbuffered binary mode: returns a FileIO
def open(
self, mode: OpenBinaryMode, buffering: Literal[0], encoding: None = ..., errors: None = ..., newline: None = ...
) -> FileIO: ...
# Buffering is on: return BufferedRandom, BufferedReader, or BufferedWriter
def open(
self,
mode: OpenBinaryModeUpdating,
buffering: Literal[-1, 1] = ...,
encoding: None = ...,
errors: None = ...,
newline: None = ...,
) -> BufferedRandom: ...
def open(
self,
mode: OpenBinaryModeWriting,
buffering: Literal[-1, 1] = ...,
encoding: None = ...,
errors: None = ...,
newline: None = ...,
) -> BufferedWriter: ...
def open(
self,
mode: OpenBinaryModeReading,
buffering: Literal[-1, 1] = ...,
encoding: None = ...,
errors: None = ...,
newline: None = ...,
) -> BufferedReader: ...
# Buffering cannot be determined: fall back to BinaryIO
def open(
self, mode: OpenBinaryMode, buffering: int, encoding: None = ..., errors: None = ..., newline: None = ...
) -> BinaryIO: ...
# Fallback if mode is not specified
def open(
self,
mode: str,
buffering: int = ...,
encoding: Optional[str] = ...,
errors: Optional[str] = ...,
newline: Optional[str] = ...,
) -> IO[Any]: ...
def owner(self) -> str: ...
if sys.version_info >= (3, 9):
def readlink(self: _P) -> _P: ...
if sys.version_info >= (3, 8):
def rename(self: _P, target: Union[str, PurePath]) -> _P: ...
def replace(self: _P, target: Union[str, PurePath]) -> _P: ...
else:
def rename(self, target: Union[str, PurePath]) -> None: ...
def replace(self, target: Union[str, PurePath]) -> None: ...
def resolve(self: _P, strict: bool = ...) -> _P: ...
def rglob(self: _P, pattern: str) -> Generator[_P, None, None]: ...
def rmdir(self) -> None: ...
def symlink_to(self, target: Union[str, Path], target_is_directory: bool = ...) -> None: ...
def touch(self, mode: int = ..., exist_ok: bool = ...) -> None: ...
if sys.version_info >= (3, 8):
def unlink(self, missing_ok: bool = ...) -> None: ...
else:
def unlink(self) -> None: ...
def home(cls: Type[_P]) -> _P: ...
def absolute(self: _P) -> _P: ...
def expanduser(self: _P) -> _P: ...
def read_bytes(self) -> bytes: ...
def read_text(self, encoding: Optional[str] = ..., errors: Optional[str] = ...) -> str: ...
def samefile(self, other_path: Union[str, bytes, int, Path]) -> bool: ...
def write_bytes(self, data: bytes) -> int: ...
def write_text(self, data: str, encoding: Optional[str] = ..., errors: Optional[str] = ...) -> int: ...
if sys.version_info >= (3, 8):
def link_to(self, target: Union[str, bytes, os.PathLike[str]]) -> None: ...
The provided code snippet includes necessary dependencies for implementing the `is_pathlib_path` function. Write a Python function `def is_pathlib_path(obj)` to solve the following problem:
Check whether obj is a `pathlib.Path` object. Prefer using ``isinstance(obj, os.PathLike)`` instead of this function.
Here is the function:
def is_pathlib_path(obj):
"""
Check whether obj is a `pathlib.Path` object.
Prefer using ``isinstance(obj, os.PathLike)`` instead of this function.
"""
return isinstance(obj, Path) | Check whether obj is a `pathlib.Path` object. Prefer using ``isinstance(obj, os.PathLike)`` instead of this function. |
169,717 | import sys
import os
from pathlib import Path
import io
The provided code snippet includes necessary dependencies for implementing the `npy_load_module` function. Write a Python function `def npy_load_module(name, fn, info=None)` to solve the following problem:
Load a module. Uses ``load_module`` which will be deprecated in python 3.12. An alternative that uses ``exec_module`` is in numpy.distutils.misc_util.exec_mod_from_location .. versionadded:: 1.11.2 Parameters ---------- name : str Full module name. fn : str Path to module file. info : tuple, optional Only here for backward compatibility with Python 2.*. Returns ------- mod : module
Here is the function:
def npy_load_module(name, fn, info=None):
"""
Load a module. Uses ``load_module`` which will be deprecated in python
3.12. An alternative that uses ``exec_module`` is in
numpy.distutils.misc_util.exec_mod_from_location
.. versionadded:: 1.11.2
Parameters
----------
name : str
Full module name.
fn : str
Path to module file.
info : tuple, optional
Only here for backward compatibility with Python 2.*.
Returns
-------
mod : module
"""
# Explicitly lazy import this to avoid paying the cost
# of importing importlib at startup
from importlib.machinery import SourceFileLoader
return SourceFileLoader(name, fn).load_module() | Load a module. Uses ``load_module`` which will be deprecated in python 3.12. An alternative that uses ``exec_module`` is in numpy.distutils.misc_util.exec_mod_from_location .. versionadded:: 1.11.2 Parameters ---------- name : str Full module name. fn : str Path to module file. info : tuple, optional Only here for backward compatibility with Python 2.*. Returns ------- mod : module |
169,718 | import collections
import itertools
import re
class InvalidVersion(ValueError):
"""
An invalid version was found, users should refer to PEP 440.
"""
class LegacyVersion(_BaseVersion):
def __init__(self, version):
self._version = str(version)
self._key = _legacy_cmpkey(self._version)
def __str__(self):
return self._version
def __repr__(self):
return "<LegacyVersion({0})>".format(repr(str(self)))
def public(self):
return self._version
def base_version(self):
return self._version
def local(self):
return None
def is_prerelease(self):
return False
def is_postrelease(self):
return False
class Version(_BaseVersion):
_regex = re.compile(
r"^\s*" + VERSION_PATTERN + r"\s*$",
re.VERBOSE | re.IGNORECASE,
)
def __init__(self, version):
# Validate the version and parse it into pieces
match = self._regex.search(version)
if not match:
raise InvalidVersion("Invalid version: '{0}'".format(version))
# Store the parsed out pieces of the version
self._version = _Version(
epoch=int(match.group("epoch")) if match.group("epoch") else 0,
release=tuple(int(i) for i in match.group("release").split(".")),
pre=_parse_letter_version(
match.group("pre_l"),
match.group("pre_n"),
),
post=_parse_letter_version(
match.group("post_l"),
match.group("post_n1") or match.group("post_n2"),
),
dev=_parse_letter_version(
match.group("dev_l"),
match.group("dev_n"),
),
local=_parse_local_version(match.group("local")),
)
# Generate a key which will be used for sorting
self._key = _cmpkey(
self._version.epoch,
self._version.release,
self._version.pre,
self._version.post,
self._version.dev,
self._version.local,
)
def __repr__(self):
return "<Version({0})>".format(repr(str(self)))
def __str__(self):
parts = []
# Epoch
if self._version.epoch != 0:
parts.append("{0}!".format(self._version.epoch))
# Release segment
parts.append(".".join(str(x) for x in self._version.release))
# Pre-release
if self._version.pre is not None:
parts.append("".join(str(x) for x in self._version.pre))
# Post-release
if self._version.post is not None:
parts.append(".post{0}".format(self._version.post[1]))
# Development release
if self._version.dev is not None:
parts.append(".dev{0}".format(self._version.dev[1]))
# Local version segment
if self._version.local is not None:
parts.append(
"+{0}".format(".".join(str(x) for x in self._version.local))
)
return "".join(parts)
def public(self):
return str(self).split("+", 1)[0]
def base_version(self):
parts = []
# Epoch
if self._version.epoch != 0:
parts.append("{0}!".format(self._version.epoch))
# Release segment
parts.append(".".join(str(x) for x in self._version.release))
return "".join(parts)
def local(self):
version_string = str(self)
if "+" in version_string:
return version_string.split("+", 1)[1]
def is_prerelease(self):
return bool(self._version.dev or self._version.pre)
def is_postrelease(self):
return bool(self._version.post)
The provided code snippet includes necessary dependencies for implementing the `parse` function. Write a Python function `def parse(version)` to solve the following problem:
Parse the given version string and return either a :class:`Version` object or a :class:`LegacyVersion` object depending on if the given version is a valid PEP 440 version or a legacy version.
Here is the function:
def parse(version):
"""
Parse the given version string and return either a :class:`Version` object
or a :class:`LegacyVersion` object depending on if the given version is
a valid PEP 440 version or a legacy version.
"""
try:
return Version(version)
except InvalidVersion:
return LegacyVersion(version) | Parse the given version string and return either a :class:`Version` object or a :class:`LegacyVersion` object depending on if the given version is a valid PEP 440 version or a legacy version. |
169,719 | import collections
import itertools
import re
def _parse_version_parts(s):
for part in _legacy_version_component_re.split(s):
part = _legacy_version_replacement_map.get(part, part)
if not part or part == ".":
continue
if part[:1] in "0123456789":
# pad for numeric comparison
yield part.zfill(8)
else:
yield "*" + part
# ensure that alpha/beta/candidate are before final
yield "*final"
def _legacy_cmpkey(version):
# We hardcode an epoch of -1 here. A PEP 440 version can only have an epoch
# greater than or equal to 0. This will effectively put the LegacyVersion,
# which uses the defacto standard originally implemented by setuptools,
# as before all PEP 440 versions.
epoch = -1
# This scheme is taken from pkg_resources.parse_version setuptools prior to
# its adoption of the packaging library.
parts = []
for part in _parse_version_parts(version.lower()):
if part.startswith("*"):
# remove "-" before a prerelease tag
if part < "*final":
while parts and parts[-1] == "*final-":
parts.pop()
# remove trailing zeros from each series of numeric parts
while parts and parts[-1] == "00000000":
parts.pop()
parts.append(part)
parts = tuple(parts)
return epoch, parts | null |
169,720 | import collections
import itertools
import re
def _parse_letter_version(letter, number):
if letter:
# We assume there is an implicit 0 in a pre-release if there is
# no numeral associated with it.
if number is None:
number = 0
# We normalize any letters to their lower-case form
letter = letter.lower()
# We consider some words to be alternate spellings of other words and
# in those cases we want to normalize the spellings to our preferred
# spelling.
if letter == "alpha":
letter = "a"
elif letter == "beta":
letter = "b"
elif letter in ["c", "pre", "preview"]:
letter = "rc"
elif letter in ["rev", "r"]:
letter = "post"
return letter, int(number)
if not letter and number:
# We assume that if we are given a number but not given a letter,
# then this is using the implicit post release syntax (e.g., 1.0-1)
letter = "post"
return letter, int(number) | null |
169,721 | import collections
import itertools
import re
_local_version_seperators = re.compile(r"[\._-]")
The provided code snippet includes necessary dependencies for implementing the `_parse_local_version` function. Write a Python function `def _parse_local_version(local)` to solve the following problem:
Takes a string like abc.1.twelve and turns it into ("abc", 1, "twelve").
Here is the function:
def _parse_local_version(local):
"""
Takes a string like abc.1.twelve and turns it into ("abc", 1, "twelve").
"""
if local is not None:
return tuple(
part.lower() if not part.isdigit() else int(part)
for part in _local_version_seperators.split(local)
) | Takes a string like abc.1.twelve and turns it into ("abc", 1, "twelve"). |
169,722 | import collections
import itertools
import re
class Infinity:
def __repr__(self):
return "Infinity"
def __hash__(self):
return hash(repr(self))
def __lt__(self, other):
return False
def __le__(self, other):
return False
def __eq__(self, other):
return isinstance(other, self.__class__)
def __ne__(self, other):
return not isinstance(other, self.__class__)
def __gt__(self, other):
return True
def __ge__(self, other):
return True
def __neg__(self):
return NegativeInfinity
Infinity = Infinity()
def _cmpkey(epoch, release, pre, post, dev, local):
# When we compare a release version, we want to compare it with all of the
# trailing zeros removed. So we'll use a reverse the list, drop all the now
# leading zeros until we come to something non-zero, then take the rest,
# re-reverse it back into the correct order, and make it a tuple and use
# that for our sorting key.
release = tuple(
reversed(list(
itertools.dropwhile(
lambda x: x == 0,
reversed(release),
)
))
)
# We need to "trick" the sorting algorithm to put 1.0.dev0 before 1.0a0.
# We'll do this by abusing the pre-segment, but we _only_ want to do this
# if there is no pre- or a post-segment. If we have one of those, then
# the normal sorting rules will handle this case correctly.
if pre is None and post is None and dev is not None:
pre = -Infinity
# Versions without a pre-release (except as noted above) should sort after
# those with one.
elif pre is None:
pre = Infinity
# Versions without a post-segment should sort before those with one.
if post is None:
post = -Infinity
# Versions without a development segment should sort after those with one.
if dev is None:
dev = Infinity
if local is None:
# Versions without a local segment should sort before those with one.
local = -Infinity
else:
# Versions with a local segment need that segment parsed to implement
# the sorting rules in PEP440.
# - Alphanumeric segments sort before numeric segments
# - Alphanumeric segments sort lexicographically
# - Numeric segments sort numerically
# - Shorter versions sort before longer versions when the prefixes
# match exactly
local = tuple(
(i, "") if isinstance(i, int) else (-Infinity, i)
for i in local
)
return epoch, release, pre, post, dev, local | null |
169,723 | import warnings
__all__ = ['fft', 'ifft', 'fftn', 'ifftn', 'fft2', 'ifft2',
'norm', 'inv', 'svd', 'solve', 'det', 'eig', 'eigvals',
'eigh', 'eigvalsh', 'lstsq', 'pinv', 'cholesky', 'i0']
import numpy.linalg as linpkg
import numpy.fft as fftpkg
from numpy.lib import i0
import sys
_restore_dict = {}
import sys
del sys, warnings
def register_func(name, func):
if name not in __all__:
raise ValueError("{} not a dual function.".format(name))
f = sys._getframe(0).f_globals
_restore_dict[name] = f[name]
f[name] = func | null |
169,724 | import warnings
import numpy.linalg as linpkg
import numpy.fft as fftpkg
from numpy.lib import i0
import sys
_restore_dict = {}
def restore_func(name):
if name not in __all__:
raise ValueError("{} not a dual function.".format(name))
try:
val = _restore_dict[name]
except KeyError:
return
else:
sys._getframe(0).f_globals[name] = val
def restore_all():
for name in _restore_dict.keys():
restore_func(name) | null |
169,725 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legadd(c1, c2):
"""
Add one Legendre series to another.
Returns the sum of two Legendre series `c1` + `c2`. The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the Legendre series of their sum.
See Also
--------
legsub, legmulx, legmul, legdiv, legpow
Notes
-----
Unlike multiplication, division, etc., the sum of two Legendre series
is a Legendre series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legadd(c1,c2)
array([4., 4., 4.])
"""
return pu._add(c1, c2)
def legmulx(c):
"""Multiply a Legendre series by x.
Multiply the Legendre series `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
See Also
--------
legadd, legmul, legdiv, legpow
Notes
-----
The multiplication uses the recursion relationship for Legendre
polynomials in the form
.. math::
xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1)
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> L.legmulx([1,2,3])
array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]*0
prd[1] = c[0]
for i in range(1, len(c)):
j = i + 1
k = i - 1
s = i + j
prd[j] = (c[i]*j)/s
prd[k] += (c[i]*i)/s
return prd
The provided code snippet includes necessary dependencies for implementing the `poly2leg` function. Write a Python function `def poly2leg(pol)` to solve the following problem:
Convert a polynomial to a Legendre series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Legendre series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Legendre series. See Also -------- leg2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy import polynomial as P >>> p = P.Polynomial(np.arange(4)) >>> p Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) >>> c = P.Legendre(P.legendre.poly2leg(p.coef)) >>> c Legendre([ 1. , 3.25, 1. , 0.75], domain=[-1, 1], window=[-1, 1]) # may vary
Here is the function:
def poly2leg(pol):
"""
Convert a polynomial to a Legendre series.
Convert an array representing the coefficients of a polynomial (relative
to the "standard" basis) ordered from lowest degree to highest, to an
array of the coefficients of the equivalent Legendre series, ordered
from lowest to highest degree.
Parameters
----------
pol : array_like
1-D array containing the polynomial coefficients
Returns
-------
c : ndarray
1-D array containing the coefficients of the equivalent Legendre
series.
See Also
--------
leg2poly
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy import polynomial as P
>>> p = P.Polynomial(np.arange(4))
>>> p
Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
>>> c = P.Legendre(P.legendre.poly2leg(p.coef))
>>> c
Legendre([ 1. , 3.25, 1. , 0.75], domain=[-1, 1], window=[-1, 1]) # may vary
"""
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
for i in range(deg, -1, -1):
res = legadd(legmulx(res), pol[i])
return res | Convert a polynomial to a Legendre series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Legendre series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Legendre series. See Also -------- leg2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy import polynomial as P >>> p = P.Polynomial(np.arange(4)) >>> p Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) >>> c = P.Legendre(P.legendre.poly2leg(p.coef)) >>> c Legendre([ 1. , 3.25, 1. , 0.75], domain=[-1, 1], window=[-1, 1]) # may vary |
169,726 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyadd(c1, c2):
"""
Add one polynomial to another.
Returns the sum of two polynomials `c1` + `c2`. The arguments are
sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to high.
Returns
-------
out : ndarray
The coefficient array representing their sum.
See Also
--------
polysub, polymulx, polymul, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> sum = P.polyadd(c1,c2); sum
array([4., 4., 4.])
>>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2)
28.0
"""
return pu._add(c1, c2)
def polysub(c1, c2):
"""
Subtract one polynomial from another.
Returns the difference of two polynomials `c1` - `c2`. The arguments
are sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of coefficients representing their difference.
See Also
--------
polyadd, polymulx, polymul, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polysub(c1,c2)
array([-2., 0., 2.])
>>> P.polysub(c2,c1) # -P.polysub(c1,c2)
array([ 2., 0., -2.])
"""
return pu._sub(c1, c2)
def polymulx(c):
"""Multiply a polynomial by x.
Multiply the polynomial `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
See Also
--------
polyadd, polysub, polymul, polydiv, polypow
Notes
-----
.. versionadded:: 1.5.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]*0
prd[1:] = c
return prd
The provided code snippet includes necessary dependencies for implementing the `leg2poly` function. Write a Python function `def leg2poly(c)` to solve the following problem:
Convert a Legendre series to a polynomial. Convert an array representing the coefficients of a Legendre series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- c : array_like 1-D array containing the Legendre series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2leg Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy import polynomial as P >>> c = P.Legendre(range(4)) >>> c Legendre([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) >>> p = c.convert(kind=P.Polynomial) >>> p Polynomial([-1. , -3.5, 3. , 7.5], domain=[-1., 1.], window=[-1., 1.]) >>> P.legendre.leg2poly(range(4)) array([-1. , -3.5, 3. , 7.5])
Here is the function:
def leg2poly(c):
"""
Convert a Legendre series to a polynomial.
Convert an array representing the coefficients of a Legendre series,
ordered from lowest degree to highest, to an array of the coefficients
of the equivalent polynomial (relative to the "standard" basis) ordered
from lowest to highest degree.
Parameters
----------
c : array_like
1-D array containing the Legendre series coefficients, ordered
from lowest order term to highest.
Returns
-------
pol : ndarray
1-D array containing the coefficients of the equivalent polynomial
(relative to the "standard" basis) ordered from lowest order term
to highest.
See Also
--------
poly2leg
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy import polynomial as P
>>> c = P.Legendre(range(4))
>>> c
Legendre([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
>>> p = c.convert(kind=P.Polynomial)
>>> p
Polynomial([-1. , -3.5, 3. , 7.5], domain=[-1., 1.], window=[-1., 1.])
>>> P.legendre.leg2poly(range(4))
array([-1. , -3.5, 3. , 7.5])
"""
from .polynomial import polyadd, polysub, polymulx
[c] = pu.as_series([c])
n = len(c)
if n < 3:
return c
else:
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], (c1*(i - 1))/i)
c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i)
return polyadd(c0, polymulx(c1)) | Convert a Legendre series to a polynomial. Convert an array representing the coefficients of a Legendre series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- c : array_like 1-D array containing the Legendre series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2leg Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy import polynomial as P >>> c = P.Legendre(range(4)) >>> c Legendre([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) >>> p = c.convert(kind=P.Polynomial) >>> p Polynomial([-1. , -3.5, 3. , 7.5], domain=[-1., 1.], window=[-1., 1.]) >>> P.legendre.leg2poly(range(4)) array([-1. , -3.5, 3. , 7.5]) |
169,727 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legline(off, scl):
"""
Legendre series whose graph is a straight line.
Parameters
----------
off, scl : scalars
The specified line is given by ``off + scl*x``.
Returns
-------
y : ndarray
This module's representation of the Legendre series for
``off + scl*x``.
See Also
--------
numpy.polynomial.polynomial.polyline
numpy.polynomial.chebyshev.chebline
numpy.polynomial.laguerre.lagline
numpy.polynomial.hermite.hermline
numpy.polynomial.hermite_e.hermeline
Examples
--------
>>> import numpy.polynomial.legendre as L
>>> L.legline(3,2)
array([3, 2])
>>> L.legval(-3, L.legline(3,2)) # should be -3
-3.0
"""
if scl != 0:
return np.array([off, scl])
else:
return np.array([off])
def legmul(c1, c2):
"""
Multiply one Legendre series by another.
Returns the product of two Legendre series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Legendre series coefficients representing their product.
See Also
--------
legadd, legsub, legmulx, legdiv, legpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Legendre polynomial basis set. Thus, to express
the product as a Legendre series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2)
>>> L.legmul(c1,c2) # multiplication requires "reprojection"
array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) # may vary
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = legsub(c[-i]*xs, (c1*(nd - 1))/nd)
c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd)
return legadd(c0, legmulx(c1))
The provided code snippet includes necessary dependencies for implementing the `legfromroots` function. Write a Python function `def legfromroots(roots)` to solve the following problem:
Generate a Legendre series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in Legendre form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x) The coefficient of the last term is not generally 1 for monic polynomials in Legendre form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below). See Also -------- numpy.polynomial.polynomial.polyfromroots numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.laguerre.lagfromroots numpy.polynomial.hermite.hermfromroots numpy.polynomial.hermite_e.hermefromroots Examples -------- >>> import numpy.polynomial.legendre as L >>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis array([ 0. , -0.4, 0. , 0.4]) >>> j = complex(0,1) >>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) # may vary
Here is the function:
def legfromroots(roots):
"""
Generate a Legendre series with given roots.
The function returns the coefficients of the polynomial
.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
in Legendre form, where the `r_n` are the roots specified in `roots`.
If a zero has multiplicity n, then it must appear in `roots` n times.
For instance, if 2 is a root of multiplicity three and 3 is a root of
multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
roots can appear in any order.
If the returned coefficients are `c`, then
.. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x)
The coefficient of the last term is not generally 1 for monic
polynomials in Legendre form.
Parameters
----------
roots : array_like
Sequence containing the roots.
Returns
-------
out : ndarray
1-D array of coefficients. If all roots are real then `out` is a
real array, if some of the roots are complex, then `out` is complex
even if all the coefficients in the result are real (see Examples
below).
See Also
--------
numpy.polynomial.polynomial.polyfromroots
numpy.polynomial.chebyshev.chebfromroots
numpy.polynomial.laguerre.lagfromroots
numpy.polynomial.hermite.hermfromroots
numpy.polynomial.hermite_e.hermefromroots
Examples
--------
>>> import numpy.polynomial.legendre as L
>>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis
array([ 0. , -0.4, 0. , 0.4])
>>> j = complex(0,1)
>>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis
array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) # may vary
"""
return pu._fromroots(legline, legmul, roots) | Generate a Legendre series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in Legendre form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x) The coefficient of the last term is not generally 1 for monic polynomials in Legendre form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below). See Also -------- numpy.polynomial.polynomial.polyfromroots numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.laguerre.lagfromroots numpy.polynomial.hermite.hermfromroots numpy.polynomial.hermite_e.hermefromroots Examples -------- >>> import numpy.polynomial.legendre as L >>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis array([ 0. , -0.4, 0. , 0.4]) >>> j = complex(0,1) >>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) # may vary |
169,728 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legmul(c1, c2):
"""
Multiply one Legendre series by another.
Returns the product of two Legendre series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Legendre series coefficients representing their product.
See Also
--------
legadd, legsub, legmulx, legdiv, legpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Legendre polynomial basis set. Thus, to express
the product as a Legendre series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2)
>>> L.legmul(c1,c2) # multiplication requires "reprojection"
array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) # may vary
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = legsub(c[-i]*xs, (c1*(nd - 1))/nd)
c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd)
return legadd(c0, legmulx(c1))
The provided code snippet includes necessary dependencies for implementing the `legdiv` function. Write a Python function `def legdiv(c1, c2)` to solve the following problem:
Divide one Legendre series by another. Returns the quotient-with-remainder of two Legendre series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Legendre series coefficients ordered from low to high. Returns ------- quo, rem : ndarrays Of Legendre series coefficients representing the quotient and remainder. See Also -------- legadd, legsub, legmulx, legmul, legpow Notes ----- In general, the (polynomial) division of one Legendre series by another results in quotient and remainder terms that are not in the Legendre polynomial basis set. Thus, to express these results as a Legendre series, it is necessary to "reproject" the results onto the Legendre basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legdiv(c1,c2) # quotient "intuitive," remainder not (array([3.]), array([-8., -4.])) >>> c2 = (0,1,2,3) >>> L.legdiv(c2,c1) # neither "intuitive" (array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) # may vary
Here is the function:
def legdiv(c1, c2):
"""
Divide one Legendre series by another.
Returns the quotient-with-remainder of two Legendre series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
quo, rem : ndarrays
Of Legendre series coefficients representing the quotient and
remainder.
See Also
--------
legadd, legsub, legmulx, legmul, legpow
Notes
-----
In general, the (polynomial) division of one Legendre series by another
results in quotient and remainder terms that are not in the Legendre
polynomial basis set. Thus, to express these results as a Legendre
series, it is necessary to "reproject" the results onto the Legendre
basis set, which may produce "unintuitive" (but correct) results; see
Examples section below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> L.legdiv(c1,c2) # quotient "intuitive," remainder not
(array([3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> L.legdiv(c2,c1) # neither "intuitive"
(array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) # may vary
"""
return pu._div(legmul, c1, c2) | Divide one Legendre series by another. Returns the quotient-with-remainder of two Legendre series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Legendre series coefficients ordered from low to high. Returns ------- quo, rem : ndarrays Of Legendre series coefficients representing the quotient and remainder. See Also -------- legadd, legsub, legmulx, legmul, legpow Notes ----- In general, the (polynomial) division of one Legendre series by another results in quotient and remainder terms that are not in the Legendre polynomial basis set. Thus, to express these results as a Legendre series, it is necessary to "reproject" the results onto the Legendre basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legdiv(c1,c2) # quotient "intuitive," remainder not (array([3.]), array([-8., -4.])) >>> c2 = (0,1,2,3) >>> L.legdiv(c2,c1) # neither "intuitive" (array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) # may vary |
169,729 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legmul(c1, c2):
"""
Multiply one Legendre series by another.
Returns the product of two Legendre series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Legendre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Legendre series coefficients representing their product.
See Also
--------
legadd, legsub, legmulx, legdiv, legpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Legendre polynomial basis set. Thus, to express
the product as a Legendre series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c1 = (1,2,3)
>>> c2 = (3,2)
>>> L.legmul(c1,c2) # multiplication requires "reprojection"
array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) # may vary
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = legsub(c[-i]*xs, (c1*(nd - 1))/nd)
c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd)
return legadd(c0, legmulx(c1))
The provided code snippet includes necessary dependencies for implementing the `legpow` function. Write a Python function `def legpow(c, pow, maxpower=16)` to solve the following problem:
Raise a Legendre series to a power. Returns the Legendre series `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` Parameters ---------- c : array_like 1-D array of Legendre series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Legendre series of power. See Also -------- legadd, legsub, legmulx, legmul, legdiv
Here is the function:
def legpow(c, pow, maxpower=16):
"""Raise a Legendre series to a power.
Returns the Legendre series `c` raised to the power `pow`. The
argument `c` is a sequence of coefficients ordered from low to high.
i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.``
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to
high.
pow : integer
Power to which the series will be raised
maxpower : integer, optional
Maximum power allowed. This is mainly to limit growth of the series
to unmanageable size. Default is 16
Returns
-------
coef : ndarray
Legendre series of power.
See Also
--------
legadd, legsub, legmulx, legmul, legdiv
"""
return pu._pow(legmul, c, pow, maxpower) | Raise a Legendre series to a power. Returns the Legendre series `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` Parameters ---------- c : array_like 1-D array of Legendre series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Legendre series of power. See Also -------- legadd, legsub, legmulx, legmul, legdiv |
169,730 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legval(x, c, tensor=True):
"""
Evaluate a Legendre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
legval2d, leggrid2d, legval3d, leggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*x*(2*nd - 1))/nd
return c0 + c1*x
The provided code snippet includes necessary dependencies for implementing the `legint` function. Write a Python function `def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0)` to solve the following problem:
Integrate a Legendre series. Returns the Legendre series coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Legendre series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at ``lbnd`` is the first value in the list, the value of the second integral at ``lbnd`` is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray Legendre series coefficient array of the integral. Raises ------ ValueError If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- legder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a` - perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> c = (1,2,3) >>> L.legint(c) array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary >>> L.legint(c, 3) array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, # may vary -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) >>> L.legint(c, k=3) array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary >>> L.legint(c, lbnd=-2) array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary >>> L.legint(c, scl=2) array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) # may vary
Here is the function:
def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
"""
Integrate a Legendre series.
Returns the Legendre series coefficients `c` integrated `m` times from
`lbnd` along `axis`. At each iteration the resulting series is
**multiplied** by `scl` and an integration constant, `k`, is added.
The scaling factor is for use in a linear change of variable. ("Buyer
beware": note that, depending on what one is doing, one may want `scl`
to be the reciprocal of what one might expect; for more information,
see the Notes section below.) The argument `c` is an array of
coefficients from low to high degree along each axis, e.g., [1,2,3]
represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
Parameters
----------
c : array_like
Array of Legendre series coefficients. If c is multidimensional the
different axis correspond to different variables with the degree in
each axis given by the corresponding index.
m : int, optional
Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
Integration constant(s). The value of the first integral at
``lbnd`` is the first value in the list, the value of the second
integral at ``lbnd`` is the second value, etc. If ``k == []`` (the
default), all constants are set to zero. If ``m == 1``, a single
scalar can be given instead of a list.
lbnd : scalar, optional
The lower bound of the integral. (Default: 0)
scl : scalar, optional
Following each integration the result is *multiplied* by `scl`
before the integration constant is added. (Default: 1)
axis : int, optional
Axis over which the integral is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
S : ndarray
Legendre series coefficient array of the integral.
Raises
------
ValueError
If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
``np.ndim(scl) != 0``.
See Also
--------
legder
Notes
-----
Note that the result of each integration is *multiplied* by `scl`.
Why is this important to note? Say one is making a linear change of
variable :math:`u = ax + b` in an integral relative to `x`. Then
:math:`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a` - perhaps not what one would have first thought.
Also note that, in general, the result of integrating a C-series needs
to be "reprojected" onto the C-series basis set. Thus, typically,
the result of this function is "unintuitive," albeit correct; see
Examples section below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c = (1,2,3)
>>> L.legint(c)
array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary
>>> L.legint(c, 3)
array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, # may vary
-1.73472348e-18, 1.90476190e-02, 9.52380952e-03])
>>> L.legint(c, k=3)
array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary
>>> L.legint(c, lbnd=-2)
array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary
>>> L.legint(c, scl=2)
array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) # may vary
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if not np.iterable(k):
k = [k]
cnt = pu._deprecate_as_int(m, "the order of integration")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of integration must be non-negative")
if len(k) > cnt:
raise ValueError("Too many integration constants")
if np.ndim(lbnd) != 0:
raise ValueError("lbnd must be a scalar.")
if np.ndim(scl) != 0:
raise ValueError("scl must be a scalar.")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
k = list(k) + [0]*(cnt - len(k))
for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
c[0] += k[i]
else:
tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
tmp[0] = c[0]*0
tmp[1] = c[0]
if n > 1:
tmp[2] = c[1]/3
for j in range(2, n):
t = c[j]/(2*j + 1)
tmp[j + 1] = t
tmp[j - 1] -= t
tmp[0] += k[i] - legval(lbnd, tmp)
c = tmp
c = np.moveaxis(c, 0, iaxis)
return c | Integrate a Legendre series. Returns the Legendre series coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Legendre series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at ``lbnd`` is the first value in the list, the value of the second integral at ``lbnd`` is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray Legendre series coefficient array of the integral. Raises ------ ValueError If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- legder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a` - perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> c = (1,2,3) >>> L.legint(c) array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary >>> L.legint(c, 3) array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, # may vary -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) >>> L.legint(c, k=3) array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary >>> L.legint(c, lbnd=-2) array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary >>> L.legint(c, scl=2) array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) # may vary |
169,731 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legval(x, c, tensor=True):
"""
Evaluate a Legendre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
legval2d, leggrid2d, legval3d, leggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*x*(2*nd - 1))/nd
return c0 + c1*x
The provided code snippet includes necessary dependencies for implementing the `legval2d` function. Write a Python function `def legval2d(x, y, c)` to solve the following problem:
Evaluate a 2-D Legendre series at points (x, y). This function returns the values: .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y) The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Legendre series at points formed from pairs of corresponding values from `x` and `y`. See Also -------- legval, leggrid2d, legval3d, leggrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def legval2d(x, y, c):
"""
Evaluate a 2-D Legendre series at points (x, y).
This function returns the values:
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars and they
must have the same shape after conversion. In either case, either `x`
and `y` or their elements must support multiplication and addition both
with themselves and with the elements of `c`.
If `c` is a 1-D array a one is implicitly appended to its shape to make
it 2-D. The shape of the result will be c.shape[2:] + x.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points `(x, y)`,
where `x` and `y` must have the same shape. If `x` or `y` is a list
or tuple, it is first converted to an ndarray, otherwise it is left
unchanged and if it isn't an ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term
of multi-degree i,j is contained in ``c[i,j]``. If `c` has
dimension greater than two the remaining indices enumerate multiple
sets of coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional Legendre series at points formed
from pairs of corresponding values from `x` and `y`.
See Also
--------
legval, leggrid2d, legval3d, leggrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(legval, c, x, y) | Evaluate a 2-D Legendre series at points (x, y). This function returns the values: .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y) The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Legendre series at points formed from pairs of corresponding values from `x` and `y`. See Also -------- legval, leggrid2d, legval3d, leggrid3d Notes ----- .. versionadded:: 1.7.0 |
169,732 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legval(x, c, tensor=True):
"""
Evaluate a Legendre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
legval2d, leggrid2d, legval3d, leggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*x*(2*nd - 1))/nd
return c0 + c1*x
The provided code snippet includes necessary dependencies for implementing the `leggrid2d` function. Write a Python function `def leggrid2d(x, y, c)` to solve the following problem:
Evaluate a 2-D Legendre series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b) where the points `(a, b)` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Chebyshev series at points in the Cartesian product of `x` and `y`. See Also -------- legval, legval2d, legval3d, leggrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def leggrid2d(x, y, c):
"""
Evaluate a 2-D Legendre series on the Cartesian product of x and y.
This function returns the values:
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)
where the points `(a, b)` consist of all pairs formed by taking
`a` from `x` and `b` from `y`. The resulting points form a grid with
`x` in the first dimension and `y` in the second.
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars. In either
case, either `x` and `y` or their elements must support multiplication
and addition both with themselves and with the elements of `c`.
If `c` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape + y.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points in the
Cartesian product of `x` and `y`. If `x` or `y` is a list or
tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional Chebyshev series at points in the
Cartesian product of `x` and `y`.
See Also
--------
legval, legval2d, legval3d, leggrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(legval, c, x, y) | Evaluate a 2-D Legendre series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b) where the points `(a, b)` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Chebyshev series at points in the Cartesian product of `x` and `y`. See Also -------- legval, legval2d, legval3d, leggrid3d Notes ----- .. versionadded:: 1.7.0 |
169,733 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legval(x, c, tensor=True):
"""
Evaluate a Legendre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
legval2d, leggrid2d, legval3d, leggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*x*(2*nd - 1))/nd
return c0 + c1*x
The provided code snippet includes necessary dependencies for implementing the `legval3d` function. Write a Python function `def legval3d(x, y, z, c)` to solve the following problem:
Evaluate a 3-D Legendre series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z) The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- legval, legval2d, leggrid2d, leggrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def legval3d(x, y, z, c):
"""
Evaluate a 3-D Legendre series at points (x, y, z).
This function returns the values:
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)
The parameters `x`, `y`, and `z` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
`x`, `y`, and `z` or their elements must support multiplication and
addition both with themselves and with the elements of `c`.
If `c` has fewer than 3 dimensions, ones are implicitly appended to its
shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape.
Parameters
----------
x, y, z : array_like, compatible object
The three dimensional series is evaluated at the points
`(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If
any of `x`, `y`, or `z` is a list or tuple, it is first converted
to an ndarray, otherwise it is left unchanged and if it isn't an
ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
greater than 3 the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the multidimensional polynomial on points formed with
triples of corresponding values from `x`, `y`, and `z`.
See Also
--------
legval, legval2d, leggrid2d, leggrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(legval, c, x, y, z) | Evaluate a 3-D Legendre series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z) The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- legval, legval2d, leggrid2d, leggrid3d Notes ----- .. versionadded:: 1.7.0 |
169,734 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legval(x, c, tensor=True):
"""
Evaluate a Legendre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
legval2d, leggrid2d, legval3d, leggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*x*(2*nd - 1))/nd
return c0 + c1*x
The provided code snippet includes necessary dependencies for implementing the `leggrid3d` function. Write a Python function `def leggrid3d(x, y, z, c)` to solve the following problem:
Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c) where the points `(a, b, c)` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- legval, legval2d, leggrid2d, legval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def leggrid3d(x, y, z, c):
"""
Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z.
This function returns the values:
.. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c)
where the points `(a, b, c)` consist of all triples formed by taking
`a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
a grid with `x` in the first dimension, `y` in the second, and `z` in
the third.
The parameters `x`, `y`, and `z` are converted to arrays only if they
are tuples or a lists, otherwise they are treated as a scalars. In
either case, either `x`, `y`, and `z` or their elements must support
multiplication and addition both with themselves and with the elements
of `c`.
If `c` has fewer than three dimensions, ones are implicitly appended to
its shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape + y.shape + z.shape.
Parameters
----------
x, y, z : array_like, compatible objects
The three dimensional series is evaluated at the points in the
Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a
list or tuple, it is first converted to an ndarray, otherwise it is
left unchanged and, if it isn't an ndarray, it is treated as a
scalar.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree i,j are contained in ``c[i,j]``. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points in the Cartesian
product of `x` and `y`.
See Also
--------
legval, legval2d, leggrid2d, legval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(legval, c, x, y, z) | Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c) where the points `(a, b, c)` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- legval, legval2d, leggrid2d, legval3d Notes ----- .. versionadded:: 1.7.0 |
169,735 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = L_i(x)
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Legendre polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and
``legval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Legendre series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo-Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Legendre polynomial. The dtype will be the same as
the converted `x`.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
# Use forward recursion to generate the entries. This is not as accurate
# as reverse recursion in this application but it is more efficient.
v[0] = x*0 + 1
if ideg > 0:
v[1] = x
for i in range(2, ideg + 1):
v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `legvander2d` function. Write a Python function `def legvander2d(x, y, deg)` to solve the following problem:
Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y)`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y), where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of `V` index the points `(x, y)` and the last index encodes the degrees of the Legendre polynomials. If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Legendre series of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- legvander, legvander3d, legval2d, legval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def legvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y)`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y),
where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
`V` index the points `(x, y)` and the last index encodes the degrees of
the Legendre polynomials.
If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
correspond to the elements of a 2-D coefficient array `c` of shape
(xdeg + 1, ydeg + 1) in the order
.. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same
up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 2-D Legendre
series of the same degrees and sample points.
Parameters
----------
x, y : array_like
Arrays of point coordinates, all of the same shape. The dtypes
will be converted to either float64 or complex128 depending on
whether any of the elements are complex. Scalars are converted to
1-D arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg].
Returns
-------
vander2d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same
as the converted `x` and `y`.
See Also
--------
legvander, legvander3d, legval2d, legval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((legvander, legvander), (x, y), deg) | Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y)`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y), where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of `V` index the points `(x, y)` and the last index encodes the degrees of the Legendre polynomials. If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Legendre series of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- legvander, legvander3d, legval2d, legval3d Notes ----- .. versionadded:: 1.7.0 |
169,736 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = L_i(x)
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Legendre polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and
``legval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Legendre series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo-Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Legendre polynomial. The dtype will be the same as
the converted `x`.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
# Use forward recursion to generate the entries. This is not as accurate
# as reverse recursion in this application but it is more efficient.
v[0] = x*0 + 1
if ideg > 0:
v[1] = x
for i in range(2, ideg + 1):
v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `legvander3d` function. Write a Python function `def legvander3d(x, y, z, deg)` to solve the following problem:
Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z), where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading indices of `V` index the points `(x, y, z)` and the last index encodes the degrees of the Legendre polynomials. If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Legendre series of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- legvander, legvander3d, legval2d, legval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def legvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
then The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),
where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading
indices of `V` index the points `(x, y, z)` and the last index encodes
the degrees of the Legendre polynomials.
If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
of `V` correspond to the elements of a 3-D coefficient array `c` of
shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
.. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the
same up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 3-D Legendre
series of the same degrees and sample points.
Parameters
----------
x, y, z : array_like
Arrays of point coordinates, all of the same shape. The dtypes will
be converted to either float64 or complex128 depending on whether
any of the elements are complex. Scalars are converted to 1-D
arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg, z_deg].
Returns
-------
vander3d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will
be the same as the converted `x`, `y`, and `z`.
See Also
--------
legvander, legvander3d, legval2d, legval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg) | Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z), where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading indices of `V` index the points `(x, y, z)` and the last index encodes the degrees of the Legendre polynomials. If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Legendre series of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- legvander, legvander3d, legval2d, legval3d Notes ----- .. versionadded:: 1.7.0 |
169,737 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = L_i(x)
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Legendre polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and
``legval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Legendre series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo-Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Legendre polynomial. The dtype will be the same as
the converted `x`.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
# Use forward recursion to generate the entries. This is not as accurate
# as reverse recursion in this application but it is more efficient.
v[0] = x*0 + 1
if ideg > 0:
v[1] = x
for i in range(2, ideg + 1):
v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `legfit` function. Write a Python function `def legfit(x, y, deg, rcond=None, full=False, w=None)` to solve the following problem:
Least squares fit of Legendre series to data. Return the coefficients of a Legendre series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x), where `n` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the weight ``w[i]`` applies to the unsquared residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. When using inverse-variance weighting, use ``w[i] = 1/sigma(y[i])``. The default value is None. .. versionadded:: 1.5.0 Returns ------- coef : ndarray, shape (M,) or (M, K) Legendre coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`. If `deg` is specified as a list, coefficients for terms not included in the fit are set equal to zero in the returned `coef`. [residuals, rank, singular_values, rcond] : list These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the numerical rank of the scaled Vandermonde matrix - singular_values -- singular values of the scaled Vandermonde matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if ``full == False``. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- numpy.polynomial.polynomial.polyfit numpy.polynomial.chebyshev.chebfit numpy.polynomial.laguerre.lagfit numpy.polynomial.hermite.hermfit numpy.polynomial.hermite_e.hermefit legval : Evaluates a Legendre series. legvander : Vandermonde matrix of Legendre series. legweight : Legendre weight function (= 1). numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Legendre series `p` that minimizes the sum of the weighted squared errors .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, where :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the coefficients to be solved for, `w` are the weights, and `y` are the observed values. This equation is then solved using the singular value decomposition of `V`. If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Legendre series are usually better conditioned than fits using power series, but much can depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples --------
Here is the function:
def legfit(x, y, deg, rcond=None, full=False, w=None):
"""
Least squares fit of Legendre series to data.
Return the coefficients of a Legendre series of degree `deg` that is the
least squares fit to the data values `y` given at points `x`. If `y` is
1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
fits are done, one for each column of `y`, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form
.. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),
where `n` is `deg`.
Parameters
----------
x : array_like, shape (M,)
x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,) or (M, K)
y-coordinates of the sample points. Several data sets of sample
points sharing the same x-coordinates can be fitted at once by
passing in a 2D-array that contains one dataset per column.
deg : int or 1-D array_like
Degree(s) of the fitting polynomials. If `deg` is a single integer
all terms up to and including the `deg`'th term are included in the
fit. For NumPy versions >= 1.11.0 a list of integers specifying the
degrees of the terms to include may be used instead.
rcond : float, optional
Relative condition number of the fit. Singular values smaller than
this relative to the largest singular value will be ignored. The
default value is len(x)*eps, where eps is the relative precision of
the float type, about 2e-16 in most cases.
full : bool, optional
Switch determining nature of return value. When it is False (the
default) just the coefficients are returned, when True diagnostic
information from the singular value decomposition is also returned.
w : array_like, shape (`M`,), optional
Weights. If not None, the weight ``w[i]`` applies to the unsquared
residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
chosen so that the errors of the products ``w[i]*y[i]`` all have the
same variance. When using inverse-variance weighting, use
``w[i] = 1/sigma(y[i])``. The default value is None.
.. versionadded:: 1.5.0
Returns
-------
coef : ndarray, shape (M,) or (M, K)
Legendre coefficients ordered from low to high. If `y` was
2-D, the coefficients for the data in column k of `y` are in
column `k`. If `deg` is specified as a list, coefficients for
terms not included in the fit are set equal to zero in the
returned `coef`.
[residuals, rank, singular_values, rcond] : list
These values are only returned if ``full == True``
- residuals -- sum of squared residuals of the least squares fit
- rank -- the numerical rank of the scaled Vandermonde matrix
- singular_values -- singular values of the scaled Vandermonde matrix
- rcond -- value of `rcond`.
For more details, see `numpy.linalg.lstsq`.
Warns
-----
RankWarning
The rank of the coefficient matrix in the least-squares fit is
deficient. The warning is only raised if ``full == False``. The
warnings can be turned off by
>>> import warnings
>>> warnings.simplefilter('ignore', np.RankWarning)
See Also
--------
numpy.polynomial.polynomial.polyfit
numpy.polynomial.chebyshev.chebfit
numpy.polynomial.laguerre.lagfit
numpy.polynomial.hermite.hermfit
numpy.polynomial.hermite_e.hermefit
legval : Evaluates a Legendre series.
legvander : Vandermonde matrix of Legendre series.
legweight : Legendre weight function (= 1).
numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes
-----
The solution is the coefficients of the Legendre series `p` that
minimizes the sum of the weighted squared errors
.. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
where :math:`w_j` are the weights. This problem is solved by setting up
as the (typically) overdetermined matrix equation
.. math:: V(x) * c = w * y,
where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
coefficients to be solved for, `w` are the weights, and `y` are the
observed values. This equation is then solved using the singular value
decomposition of `V`.
If some of the singular values of `V` are so small that they are
neglected, then a `RankWarning` will be issued. This means that the
coefficient values may be poorly determined. Using a lower order fit
will usually get rid of the warning. The `rcond` parameter can also be
set to a value smaller than its default, but the resulting fit may be
spurious and have large contributions from roundoff error.
Fits using Legendre series are usually better conditioned than fits
using power series, but much can depend on the distribution of the
sample points and the smoothness of the data. If the quality of the fit
is inadequate splines may be a good alternative.
References
----------
.. [1] Wikipedia, "Curve fitting",
https://en.wikipedia.org/wiki/Curve_fitting
Examples
--------
"""
return pu._fit(legvander, x, y, deg, rcond, full, w) | Least squares fit of Legendre series to data. Return the coefficients of a Legendre series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x), where `n` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the weight ``w[i]`` applies to the unsquared residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. When using inverse-variance weighting, use ``w[i] = 1/sigma(y[i])``. The default value is None. .. versionadded:: 1.5.0 Returns ------- coef : ndarray, shape (M,) or (M, K) Legendre coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`. If `deg` is specified as a list, coefficients for terms not included in the fit are set equal to zero in the returned `coef`. [residuals, rank, singular_values, rcond] : list These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the numerical rank of the scaled Vandermonde matrix - singular_values -- singular values of the scaled Vandermonde matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if ``full == False``. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- numpy.polynomial.polynomial.polyfit numpy.polynomial.chebyshev.chebfit numpy.polynomial.laguerre.lagfit numpy.polynomial.hermite.hermfit numpy.polynomial.hermite_e.hermefit legval : Evaluates a Legendre series. legvander : Vandermonde matrix of Legendre series. legweight : Legendre weight function (= 1). numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Legendre series `p` that minimizes the sum of the weighted squared errors .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, where :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the coefficients to be solved for, `w` are the weights, and `y` are the observed values. This equation is then solved using the singular value decomposition of `V`. If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Legendre series are usually better conditioned than fits using power series, but much can depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples -------- |
169,738 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Legendre basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded:: 1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = 1./np.sqrt(2*np.arange(n) + 1)
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
return mat
The provided code snippet includes necessary dependencies for implementing the `legroots` function. Write a Python function `def legroots(c)` to solve the following problem:
Compute the roots of a Legendre series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * L_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex. See Also -------- numpy.polynomial.polynomial.polyroots numpy.polynomial.chebyshev.chebroots numpy.polynomial.laguerre.lagroots numpy.polynomial.hermite.hermroots numpy.polynomial.hermite_e.hermeroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method. The Legendre series basis polynomials aren't powers of ``x`` so the results of this function may seem unintuitive. Examples -------- >>> import numpy.polynomial.legendre as leg >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots array([-0.85099543, -0.11407192, 0.51506735]) # may vary
Here is the function:
def legroots(c):
"""
Compute the roots of a Legendre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
numpy.polynomial.polynomial.polyroots
numpy.polynomial.chebyshev.chebroots
numpy.polynomial.laguerre.lagroots
numpy.polynomial.hermite.hermroots
numpy.polynomial.hermite_e.hermeroots
Notes
-----
The root estimates are obtained as the eigenvalues of the companion
matrix, Roots far from the origin of the complex plane may have large
errors due to the numerical instability of the series for such values.
Roots with multiplicity greater than 1 will also show larger errors as
the value of the series near such points is relatively insensitive to
errors in the roots. Isolated roots near the origin can be improved by
a few iterations of Newton's method.
The Legendre series basis polynomials aren't powers of ``x`` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.legendre as leg
>>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
array([-0.85099543, -0.11407192, 0.51506735]) # may vary
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
# rotated companion matrix reduces error
m = legcompanion(c)[::-1,::-1]
r = la.eigvals(m)
r.sort()
return r | Compute the roots of a Legendre series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * L_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex. See Also -------- numpy.polynomial.polynomial.polyroots numpy.polynomial.chebyshev.chebroots numpy.polynomial.laguerre.lagroots numpy.polynomial.hermite.hermroots numpy.polynomial.hermite_e.hermeroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method. The Legendre series basis polynomials aren't powers of ``x`` so the results of this function may seem unintuitive. Examples -------- >>> import numpy.polynomial.legendre as leg >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots array([-0.85099543, -0.11407192, 0.51506735]) # may vary |
169,739 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def legder(c, m=1, scl=1, axis=0):
"""
Differentiate a Legendre series.
Returns the Legendre series coefficients `c` differentiated `m` times
along `axis`. At each iteration the result is multiplied by `scl` (the
scaling factor is for use in a linear change of variable). The argument
`c` is an array of coefficients from low to high degree along each
axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2``
while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
``y``.
Parameters
----------
c : array_like
Array of Legendre series coefficients. If c is multidimensional the
different axis correspond to different variables with the degree in
each axis given by the corresponding index.
m : int, optional
Number of derivatives taken, must be non-negative. (Default: 1)
scl : scalar, optional
Each differentiation is multiplied by `scl`. The end result is
multiplication by ``scl**m``. This is for use in a linear change of
variable. (Default: 1)
axis : int, optional
Axis over which the derivative is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
der : ndarray
Legendre series of the derivative.
See Also
--------
legint
Notes
-----
In general, the result of differentiating a Legendre series does not
resemble the same operation on a power series. Thus the result of this
function may be "unintuitive," albeit correct; see Examples section
below.
Examples
--------
>>> from numpy.polynomial import legendre as L
>>> c = (1,2,3,4)
>>> L.legder(c)
array([ 6., 9., 20.])
>>> L.legder(c, 3)
array([60.])
>>> L.legder(c, scl=-1)
array([ -6., -9., -20.])
>>> L.legder(c, 2,-1)
array([ 9., 60.])
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
cnt = pu._deprecate_as_int(m, "the order of derivation")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of derivation must be non-negative")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
n = len(c)
if cnt >= n:
c = c[:1]*0
else:
for i in range(cnt):
n = n - 1
c *= scl
der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
for j in range(n, 2, -1):
der[j - 1] = (2*j - 1)*c[j]
c[j - 2] += c[j]
if n > 1:
der[1] = 3*c[2]
der[0] = c[1]
c = der
c = np.moveaxis(c, 0, iaxis)
return c
def legval(x, c, tensor=True):
"""
Evaluate a Legendre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
legval2d, leggrid2d, legval3d, leggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*x*(2*nd - 1))/nd
return c0 + c1*x
def legcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Legendre basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Legendre series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded:: 1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = 1./np.sqrt(2*np.arange(n) + 1)
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
return mat
The provided code snippet includes necessary dependencies for implementing the `leggauss` function. Write a Python function `def leggauss(deg)` to solve the following problem:
Gauss-Legendre quadrature. Computes the sample points and weights for Gauss-Legendre quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with the weight function :math:`f(x) = 1`. Parameters ---------- deg : int Number of sample points and weights. It must be >= 1. Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights. Notes ----- .. versionadded:: 1.7.0 The results have only been tested up to degree 100, higher degrees may be problematic. The weights are determined by using the fact that .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k)) where :math:`c` is a constant independent of :math:`k` and :math:`x_k` is the k'th root of :math:`L_n`, and then scaling the results to get the right value when integrating 1.
Here is the function:
def leggauss(deg):
"""
Gauss-Legendre quadrature.
Computes the sample points and weights for Gauss-Legendre quadrature.
These sample points and weights will correctly integrate polynomials of
degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
the weight function :math:`f(x) = 1`.
Parameters
----------
deg : int
Number of sample points and weights. It must be >= 1.
Returns
-------
x : ndarray
1-D ndarray containing the sample points.
y : ndarray
1-D ndarray containing the weights.
Notes
-----
.. versionadded:: 1.7.0
The results have only been tested up to degree 100, higher degrees may
be problematic. The weights are determined by using the fact that
.. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k))
where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
is the k'th root of :math:`L_n`, and then scaling the results to get
the right value when integrating 1.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg <= 0:
raise ValueError("deg must be a positive integer")
# first approximation of roots. We use the fact that the companion
# matrix is symmetric in this case in order to obtain better zeros.
c = np.array([0]*deg + [1])
m = legcompanion(c)
x = la.eigvalsh(m)
# improve roots by one application of Newton
dy = legval(x, c)
df = legval(x, legder(c))
x -= dy/df
# compute the weights. We scale the factor to avoid possible numerical
# overflow.
fm = legval(x, c[1:])
fm /= np.abs(fm).max()
df /= np.abs(df).max()
w = 1/(fm * df)
# for Legendre we can also symmetrize
w = (w + w[::-1])/2
x = (x - x[::-1])/2
# scale w to get the right value
w *= 2. / w.sum()
return x, w | Gauss-Legendre quadrature. Computes the sample points and weights for Gauss-Legendre quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with the weight function :math:`f(x) = 1`. Parameters ---------- deg : int Number of sample points and weights. It must be >= 1. Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights. Notes ----- .. versionadded:: 1.7.0 The results have only been tested up to degree 100, higher degrees may be problematic. The weights are determined by using the fact that .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k)) where :math:`c` is a constant independent of :math:`k` and :math:`x_k` is the k'th root of :math:`L_n`, and then scaling the results to get the right value when integrating 1. |
169,740 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `legweight` function. Write a Python function `def legweight(x)` to solve the following problem:
Weight function of the Legendre polynomials. The weight function is :math:`1` and the interval of integration is :math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not normalized, with respect to this weight function. Parameters ---------- x : array_like Values at which the weight function will be computed. Returns ------- w : ndarray The weight function at `x`. Notes ----- .. versionadded:: 1.7.0
Here is the function:
def legweight(x):
"""
Weight function of the Legendre polynomials.
The weight function is :math:`1` and the interval of integration is
:math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not
normalized, with respect to this weight function.
Parameters
----------
x : array_like
Values at which the weight function will be computed.
Returns
-------
w : ndarray
The weight function at `x`.
Notes
-----
.. versionadded:: 1.7.0
"""
w = x*0.0 + 1.0
return w | Weight function of the Legendre polynomials. The weight function is :math:`1` and the interval of integration is :math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not normalized, with respect to this weight function. Parameters ---------- x : array_like Values at which the weight function will be computed. Returns ------- w : ndarray The weight function at `x`. Notes ----- .. versionadded:: 1.7.0 |
169,741 |
class Configuration:
_list_keys = ['packages', 'ext_modules', 'data_files', 'include_dirs',
'libraries', 'headers', 'scripts', 'py_modules',
'installed_libraries', 'define_macros']
_dict_keys = ['package_dir', 'installed_pkg_config']
_extra_keys = ['name', 'version']
numpy_include_dirs = []
def __init__(self,
package_name=None,
parent_name=None,
top_path=None,
package_path=None,
caller_level=1,
setup_name='setup.py',
**attrs):
"""Construct configuration instance of a package.
package_name -- name of the package
Ex.: 'distutils'
parent_name -- name of the parent package
Ex.: 'numpy'
top_path -- directory of the toplevel package
Ex.: the directory where the numpy package source sits
package_path -- directory of package. Will be computed by magic from the
directory of the caller module if not specified
Ex.: the directory where numpy.distutils is
caller_level -- frame level to caller namespace, internal parameter.
"""
self.name = dot_join(parent_name, package_name)
self.version = None
caller_frame = get_frame(caller_level)
self.local_path = get_path_from_frame(caller_frame, top_path)
# local_path -- directory of a file (usually setup.py) that
# defines a configuration() function.
# local_path -- directory of a file (usually setup.py) that
# defines a configuration() function.
if top_path is None:
top_path = self.local_path
self.local_path = ''
if package_path is None:
package_path = self.local_path
elif os.path.isdir(njoin(self.local_path, package_path)):
package_path = njoin(self.local_path, package_path)
if not os.path.isdir(package_path or '.'):
raise ValueError("%r is not a directory" % (package_path,))
self.top_path = top_path
self.package_path = package_path
# this is the relative path in the installed package
self.path_in_package = os.path.join(*self.name.split('.'))
self.list_keys = self._list_keys[:]
self.dict_keys = self._dict_keys[:]
for n in self.list_keys:
v = copy.copy(attrs.get(n, []))
setattr(self, n, as_list(v))
for n in self.dict_keys:
v = copy.copy(attrs.get(n, {}))
setattr(self, n, v)
known_keys = self.list_keys + self.dict_keys
self.extra_keys = self._extra_keys[:]
for n in attrs.keys():
if n in known_keys:
continue
a = attrs[n]
setattr(self, n, a)
if isinstance(a, list):
self.list_keys.append(n)
elif isinstance(a, dict):
self.dict_keys.append(n)
else:
self.extra_keys.append(n)
if os.path.exists(njoin(package_path, '__init__.py')):
self.packages.append(self.name)
self.package_dir[self.name] = package_path
self.options = dict(
ignore_setup_xxx_py = False,
assume_default_configuration = False,
delegate_options_to_subpackages = False,
quiet = False,
)
caller_instance = None
for i in range(1, 3):
try:
f = get_frame(i)
except ValueError:
break
try:
caller_instance = eval('self', f.f_globals, f.f_locals)
break
except NameError:
pass
if isinstance(caller_instance, self.__class__):
if caller_instance.options['delegate_options_to_subpackages']:
self.set_options(**caller_instance.options)
self.setup_name = setup_name
def todict(self):
"""
Return a dictionary compatible with the keyword arguments of distutils
setup function.
Examples
--------
>>> setup(**config.todict()) #doctest: +SKIP
"""
self._optimize_data_files()
d = {}
known_keys = self.list_keys + self.dict_keys + self.extra_keys
for n in known_keys:
a = getattr(self, n)
if a:
d[n] = a
return d
def info(self, message):
if not self.options['quiet']:
print(message)
def warn(self, message):
sys.stderr.write('Warning: %s\n' % (message,))
def set_options(self, **options):
"""
Configure Configuration instance.
The following options are available:
- ignore_setup_xxx_py
- assume_default_configuration
- delegate_options_to_subpackages
- quiet
"""
for key, value in options.items():
if key in self.options:
self.options[key] = value
else:
raise ValueError('Unknown option: '+key)
def get_distribution(self):
"""Return the distutils distribution object for self."""
from numpy.distutils.core import get_distribution
return get_distribution()
def _wildcard_get_subpackage(self, subpackage_name,
parent_name,
caller_level = 1):
l = subpackage_name.split('.')
subpackage_path = njoin([self.local_path]+l)
dirs = [_m for _m in sorted_glob(subpackage_path) if os.path.isdir(_m)]
config_list = []
for d in dirs:
if not os.path.isfile(njoin(d, '__init__.py')):
continue
if 'build' in d.split(os.sep):
continue
n = '.'.join(d.split(os.sep)[-len(l):])
c = self.get_subpackage(n,
parent_name = parent_name,
caller_level = caller_level+1)
config_list.extend(c)
return config_list
def _get_configuration_from_setup_py(self, setup_py,
subpackage_name,
subpackage_path,
parent_name,
caller_level = 1):
# In case setup_py imports local modules:
sys.path.insert(0, os.path.dirname(setup_py))
try:
setup_name = os.path.splitext(os.path.basename(setup_py))[0]
n = dot_join(self.name, subpackage_name, setup_name)
setup_module = exec_mod_from_location(
'_'.join(n.split('.')), setup_py)
if not hasattr(setup_module, 'configuration'):
if not self.options['assume_default_configuration']:
self.warn('Assuming default configuration '\
'(%s does not define configuration())'\
% (setup_module))
config = Configuration(subpackage_name, parent_name,
self.top_path, subpackage_path,
caller_level = caller_level + 1)
else:
pn = dot_join(*([parent_name] + subpackage_name.split('.')[:-1]))
args = (pn,)
if setup_module.configuration.__code__.co_argcount > 1:
args = args + (self.top_path,)
config = setup_module.configuration(*args)
if config.name!=dot_join(parent_name, subpackage_name):
self.warn('Subpackage %r configuration returned as %r' % \
(dot_join(parent_name, subpackage_name), config.name))
finally:
del sys.path[0]
return config
def get_subpackage(self,subpackage_name,
subpackage_path=None,
parent_name=None,
caller_level = 1):
"""Return list of subpackage configurations.
Parameters
----------
subpackage_name : str or None
Name of the subpackage to get the configuration. '*' in
subpackage_name is handled as a wildcard.
subpackage_path : str
If None, then the path is assumed to be the local path plus the
subpackage_name. If a setup.py file is not found in the
subpackage_path, then a default configuration is used.
parent_name : str
Parent name.
"""
if subpackage_name is None:
if subpackage_path is None:
raise ValueError(
"either subpackage_name or subpackage_path must be specified")
subpackage_name = os.path.basename(subpackage_path)
# handle wildcards
l = subpackage_name.split('.')
if subpackage_path is None and '*' in subpackage_name:
return self._wildcard_get_subpackage(subpackage_name,
parent_name,
caller_level = caller_level+1)
assert '*' not in subpackage_name, repr((subpackage_name, subpackage_path, parent_name))
if subpackage_path is None:
subpackage_path = njoin([self.local_path] + l)
else:
subpackage_path = njoin([subpackage_path] + l[:-1])
subpackage_path = self.paths([subpackage_path])[0]
setup_py = njoin(subpackage_path, self.setup_name)
if not self.options['ignore_setup_xxx_py']:
if not os.path.isfile(setup_py):
setup_py = njoin(subpackage_path,
'setup_%s.py' % (subpackage_name))
if not os.path.isfile(setup_py):
if not self.options['assume_default_configuration']:
self.warn('Assuming default configuration '\
'(%s/{setup_%s,setup}.py was not found)' \
% (os.path.dirname(setup_py), subpackage_name))
config = Configuration(subpackage_name, parent_name,
self.top_path, subpackage_path,
caller_level = caller_level+1)
else:
config = self._get_configuration_from_setup_py(
setup_py,
subpackage_name,
subpackage_path,
parent_name,
caller_level = caller_level + 1)
if config:
return [config]
else:
return []
def add_subpackage(self,subpackage_name,
subpackage_path=None,
standalone = False):
"""Add a sub-package to the current Configuration instance.
This is useful in a setup.py script for adding sub-packages to a
package.
Parameters
----------
subpackage_name : str
name of the subpackage
subpackage_path : str
if given, the subpackage path such as the subpackage is in
subpackage_path / subpackage_name. If None,the subpackage is
assumed to be located in the local path / subpackage_name.
standalone : bool
"""
if standalone:
parent_name = None
else:
parent_name = self.name
config_list = self.get_subpackage(subpackage_name, subpackage_path,
parent_name = parent_name,
caller_level = 2)
if not config_list:
self.warn('No configuration returned, assuming unavailable.')
for config in config_list:
d = config
if isinstance(config, Configuration):
d = config.todict()
assert isinstance(d, dict), repr(type(d))
self.info('Appending %s configuration to %s' \
% (d.get('name'), self.name))
self.dict_append(**d)
dist = self.get_distribution()
if dist is not None:
self.warn('distutils distribution has been initialized,'\
' it may be too late to add a subpackage '+ subpackage_name)
def add_data_dir(self, data_path):
"""Recursively add files under data_path to data_files list.
Recursively add files under data_path to the list of data_files to be
installed (and distributed). The data_path can be either a relative
path-name, or an absolute path-name, or a 2-tuple where the first
argument shows where in the install directory the data directory
should be installed to.
Parameters
----------
data_path : seq or str
Argument can be either
* 2-sequence (<datadir suffix>, <path to data directory>)
* path to data directory where python datadir suffix defaults
to package dir.
Notes
-----
Rules for installation paths::
foo/bar -> (foo/bar, foo/bar) -> parent/foo/bar
(gun, foo/bar) -> parent/gun
foo/* -> (foo/a, foo/a), (foo/b, foo/b) -> parent/foo/a, parent/foo/b
(gun, foo/*) -> (gun, foo/a), (gun, foo/b) -> gun
(gun/*, foo/*) -> parent/gun/a, parent/gun/b
/foo/bar -> (bar, /foo/bar) -> parent/bar
(gun, /foo/bar) -> parent/gun
(fun/*/gun/*, sun/foo/bar) -> parent/fun/foo/gun/bar
Examples
--------
For example suppose the source directory contains fun/foo.dat and
fun/bar/car.dat:
>>> self.add_data_dir('fun') #doctest: +SKIP
>>> self.add_data_dir(('sun', 'fun')) #doctest: +SKIP
>>> self.add_data_dir(('gun', '/full/path/to/fun'))#doctest: +SKIP
Will install data-files to the locations::
<package install directory>/
fun/
foo.dat
bar/
car.dat
sun/
foo.dat
bar/
car.dat
gun/
foo.dat
car.dat
"""
if is_sequence(data_path):
d, data_path = data_path
else:
d = None
if is_sequence(data_path):
[self.add_data_dir((d, p)) for p in data_path]
return
if not is_string(data_path):
raise TypeError("not a string: %r" % (data_path,))
if d is None:
if os.path.isabs(data_path):
return self.add_data_dir((os.path.basename(data_path), data_path))
return self.add_data_dir((data_path, data_path))
paths = self.paths(data_path, include_non_existing=False)
if is_glob_pattern(data_path):
if is_glob_pattern(d):
pattern_list = allpath(d).split(os.sep)
pattern_list.reverse()
# /a/*//b/ -> /a/*/b
rl = list(range(len(pattern_list)-1)); rl.reverse()
for i in rl:
if not pattern_list[i]:
del pattern_list[i]
#
for path in paths:
if not os.path.isdir(path):
print('Not a directory, skipping', path)
continue
rpath = rel_path(path, self.local_path)
path_list = rpath.split(os.sep)
path_list.reverse()
target_list = []
i = 0
for s in pattern_list:
if is_glob_pattern(s):
if i>=len(path_list):
raise ValueError('cannot fill pattern %r with %r' \
% (d, path))
target_list.append(path_list[i])
else:
assert s==path_list[i], repr((s, path_list[i], data_path, d, path, rpath))
target_list.append(s)
i += 1
if path_list[i:]:
self.warn('mismatch of pattern_list=%s and path_list=%s'\
% (pattern_list, path_list))
target_list.reverse()
self.add_data_dir((os.sep.join(target_list), path))
else:
for path in paths:
self.add_data_dir((d, path))
return
assert not is_glob_pattern(d), repr(d)
dist = self.get_distribution()
if dist is not None and dist.data_files is not None:
data_files = dist.data_files
else:
data_files = self.data_files
for path in paths:
for d1, f in list(general_source_directories_files(path)):
target_path = os.path.join(self.path_in_package, d, d1)
data_files.append((target_path, f))
def _optimize_data_files(self):
data_dict = {}
for p, files in self.data_files:
if p not in data_dict:
data_dict[p] = set()
for f in files:
data_dict[p].add(f)
self.data_files[:] = [(p, list(files)) for p, files in data_dict.items()]
def add_data_files(self,*files):
"""Add data files to configuration data_files.
Parameters
----------
files : sequence
Argument(s) can be either
* 2-sequence (<datadir prefix>,<path to data file(s)>)
* paths to data files where python datadir prefix defaults
to package dir.
Notes
-----
The form of each element of the files sequence is very flexible
allowing many combinations of where to get the files from the package
and where they should ultimately be installed on the system. The most
basic usage is for an element of the files argument sequence to be a
simple filename. This will cause that file from the local path to be
installed to the installation path of the self.name package (package
path). The file argument can also be a relative path in which case the
entire relative path will be installed into the package directory.
Finally, the file can be an absolute path name in which case the file
will be found at the absolute path name but installed to the package
path.
This basic behavior can be augmented by passing a 2-tuple in as the
file argument. The first element of the tuple should specify the
relative path (under the package install directory) where the
remaining sequence of files should be installed to (it has nothing to
do with the file-names in the source distribution). The second element
of the tuple is the sequence of files that should be installed. The
files in this sequence can be filenames, relative paths, or absolute
paths. For absolute paths the file will be installed in the top-level
package installation directory (regardless of the first argument).
Filenames and relative path names will be installed in the package
install directory under the path name given as the first element of
the tuple.
Rules for installation paths:
#. file.txt -> (., file.txt)-> parent/file.txt
#. foo/file.txt -> (foo, foo/file.txt) -> parent/foo/file.txt
#. /foo/bar/file.txt -> (., /foo/bar/file.txt) -> parent/file.txt
#. ``*``.txt -> parent/a.txt, parent/b.txt
#. foo/``*``.txt`` -> parent/foo/a.txt, parent/foo/b.txt
#. ``*/*.txt`` -> (``*``, ``*``/``*``.txt) -> parent/c/a.txt, parent/d/b.txt
#. (sun, file.txt) -> parent/sun/file.txt
#. (sun, bar/file.txt) -> parent/sun/file.txt
#. (sun, /foo/bar/file.txt) -> parent/sun/file.txt
#. (sun, ``*``.txt) -> parent/sun/a.txt, parent/sun/b.txt
#. (sun, bar/``*``.txt) -> parent/sun/a.txt, parent/sun/b.txt
#. (sun/``*``, ``*``/``*``.txt) -> parent/sun/c/a.txt, parent/d/b.txt
An additional feature is that the path to a data-file can actually be
a function that takes no arguments and returns the actual path(s) to
the data-files. This is useful when the data files are generated while
building the package.
Examples
--------
Add files to the list of data_files to be included with the package.
>>> self.add_data_files('foo.dat',
... ('fun', ['gun.dat', 'nun/pun.dat', '/tmp/sun.dat']),
... 'bar/cat.dat',
... '/full/path/to/can.dat') #doctest: +SKIP
will install these data files to::
<package install directory>/
foo.dat
fun/
gun.dat
nun/
pun.dat
sun.dat
bar/
car.dat
can.dat
where <package install directory> is the package (or sub-package)
directory such as '/usr/lib/python2.4/site-packages/mypackage' ('C:
\\Python2.4 \\Lib \\site-packages \\mypackage') or
'/usr/lib/python2.4/site- packages/mypackage/mysubpackage' ('C:
\\Python2.4 \\Lib \\site-packages \\mypackage \\mysubpackage').
"""
if len(files)>1:
for f in files:
self.add_data_files(f)
return
assert len(files)==1
if is_sequence(files[0]):
d, files = files[0]
else:
d = None
if is_string(files):
filepat = files
elif is_sequence(files):
if len(files)==1:
filepat = files[0]
else:
for f in files:
self.add_data_files((d, f))
return
else:
raise TypeError(repr(type(files)))
if d is None:
if hasattr(filepat, '__call__'):
d = ''
elif os.path.isabs(filepat):
d = ''
else:
d = os.path.dirname(filepat)
self.add_data_files((d, files))
return
paths = self.paths(filepat, include_non_existing=False)
if is_glob_pattern(filepat):
if is_glob_pattern(d):
pattern_list = d.split(os.sep)
pattern_list.reverse()
for path in paths:
path_list = path.split(os.sep)
path_list.reverse()
path_list.pop() # filename
target_list = []
i = 0
for s in pattern_list:
if is_glob_pattern(s):
target_list.append(path_list[i])
i += 1
else:
target_list.append(s)
target_list.reverse()
self.add_data_files((os.sep.join(target_list), path))
else:
self.add_data_files((d, paths))
return
assert not is_glob_pattern(d), repr((d, filepat))
dist = self.get_distribution()
if dist is not None and dist.data_files is not None:
data_files = dist.data_files
else:
data_files = self.data_files
data_files.append((os.path.join(self.path_in_package, d), paths))
### XXX Implement add_py_modules
def add_define_macros(self, macros):
"""Add define macros to configuration
Add the given sequence of macro name and value duples to the beginning
of the define_macros list This list will be visible to all extension
modules of the current package.
"""
dist = self.get_distribution()
if dist is not None:
if not hasattr(dist, 'define_macros'):
dist.define_macros = []
dist.define_macros.extend(macros)
else:
self.define_macros.extend(macros)
def add_include_dirs(self,*paths):
"""Add paths to configuration include directories.
Add the given sequence of paths to the beginning of the include_dirs
list. This list will be visible to all extension modules of the
current package.
"""
include_dirs = self.paths(paths)
dist = self.get_distribution()
if dist is not None:
if dist.include_dirs is None:
dist.include_dirs = []
dist.include_dirs.extend(include_dirs)
else:
self.include_dirs.extend(include_dirs)
def add_headers(self,*files):
"""Add installable headers to configuration.
Add the given sequence of files to the beginning of the headers list.
By default, headers will be installed under <python-
include>/<self.name.replace('.','/')>/ directory. If an item of files
is a tuple, then its first argument specifies the actual installation
location relative to the <python-include> path.
Parameters
----------
files : str or seq
Argument(s) can be either:
* 2-sequence (<includedir suffix>,<path to header file(s)>)
* path(s) to header file(s) where python includedir suffix will
default to package name.
"""
headers = []
for path in files:
if is_string(path):
[headers.append((self.name, p)) for p in self.paths(path)]
else:
if not isinstance(path, (tuple, list)) or len(path) != 2:
raise TypeError(repr(path))
[headers.append((path[0], p)) for p in self.paths(path[1])]
dist = self.get_distribution()
if dist is not None:
if dist.headers is None:
dist.headers = []
dist.headers.extend(headers)
else:
self.headers.extend(headers)
def paths(self,*paths,**kws):
"""Apply glob to paths and prepend local_path if needed.
Applies glob.glob(...) to each path in the sequence (if needed) and
pre-pends the local_path if needed. Because this is called on all
source lists, this allows wildcard characters to be specified in lists
of sources for extension modules and libraries and scripts and allows
path-names be relative to the source directory.
"""
include_non_existing = kws.get('include_non_existing', True)
return gpaths(paths,
local_path = self.local_path,
include_non_existing=include_non_existing)
def _fix_paths_dict(self, kw):
for k in kw.keys():
v = kw[k]
if k in ['sources', 'depends', 'include_dirs', 'library_dirs',
'module_dirs', 'extra_objects']:
new_v = self.paths(v)
kw[k] = new_v
def add_extension(self,name,sources,**kw):
"""Add extension to configuration.
Create and add an Extension instance to the ext_modules list. This
method also takes the following optional keyword arguments that are
passed on to the Extension constructor.
Parameters
----------
name : str
name of the extension
sources : seq
list of the sources. The list of sources may contain functions
(called source generators) which must take an extension instance
and a build directory as inputs and return a source file or list of
source files or None. If None is returned then no sources are
generated. If the Extension instance has no sources after
processing all source generators, then no extension module is
built.
include_dirs :
define_macros :
undef_macros :
library_dirs :
libraries :
runtime_library_dirs :
extra_objects :
extra_compile_args :
extra_link_args :
extra_f77_compile_args :
extra_f90_compile_args :
export_symbols :
swig_opts :
depends :
The depends list contains paths to files or directories that the
sources of the extension module depend on. If any path in the
depends list is newer than the extension module, then the module
will be rebuilt.
language :
f2py_options :
module_dirs :
extra_info : dict or list
dict or list of dict of keywords to be appended to keywords.
Notes
-----
The self.paths(...) method is applied to all lists that may contain
paths.
"""
ext_args = copy.copy(kw)
ext_args['name'] = dot_join(self.name, name)
ext_args['sources'] = sources
if 'extra_info' in ext_args:
extra_info = ext_args['extra_info']
del ext_args['extra_info']
if isinstance(extra_info, dict):
extra_info = [extra_info]
for info in extra_info:
assert isinstance(info, dict), repr(info)
dict_append(ext_args,**info)
self._fix_paths_dict(ext_args)
# Resolve out-of-tree dependencies
libraries = ext_args.get('libraries', [])
libnames = []
ext_args['libraries'] = []
for libname in libraries:
if isinstance(libname, tuple):
self._fix_paths_dict(libname[1])
# Handle library names of the form libname@relative/path/to/library
if '@' in libname:
lname, lpath = libname.split('@', 1)
lpath = os.path.abspath(njoin(self.local_path, lpath))
if os.path.isdir(lpath):
c = self.get_subpackage(None, lpath,
caller_level = 2)
if isinstance(c, Configuration):
c = c.todict()
for l in [l[0] for l in c.get('libraries', [])]:
llname = l.split('__OF__', 1)[0]
if llname == lname:
c.pop('name', None)
dict_append(ext_args,**c)
break
continue
libnames.append(libname)
ext_args['libraries'] = libnames + ext_args['libraries']
ext_args['define_macros'] = \
self.define_macros + ext_args.get('define_macros', [])
from numpy.distutils.core import Extension
ext = Extension(**ext_args)
self.ext_modules.append(ext)
dist = self.get_distribution()
if dist is not None:
self.warn('distutils distribution has been initialized,'\
' it may be too late to add an extension '+name)
return ext
def add_library(self,name,sources,**build_info):
"""
Add library to configuration.
Parameters
----------
name : str
Name of the extension.
sources : sequence
List of the sources. The list of sources may contain functions
(called source generators) which must take an extension instance
and a build directory as inputs and return a source file or list of
source files or None. If None is returned then no sources are
generated. If the Extension instance has no sources after
processing all source generators, then no extension module is
built.
build_info : dict, optional
The following keys are allowed:
* depends
* macros
* include_dirs
* extra_compiler_args
* extra_f77_compile_args
* extra_f90_compile_args
* f2py_options
* language
"""
self._add_library(name, sources, None, build_info)
dist = self.get_distribution()
if dist is not None:
self.warn('distutils distribution has been initialized,'\
' it may be too late to add a library '+ name)
def _add_library(self, name, sources, install_dir, build_info):
"""Common implementation for add_library and add_installed_library. Do
not use directly"""
build_info = copy.copy(build_info)
build_info['sources'] = sources
# Sometimes, depends is not set up to an empty list by default, and if
# depends is not given to add_library, distutils barfs (#1134)
if not 'depends' in build_info:
build_info['depends'] = []
self._fix_paths_dict(build_info)
# Add to libraries list so that it is build with build_clib
self.libraries.append((name, build_info))
def add_installed_library(self, name, sources, install_dir, build_info=None):
"""
Similar to add_library, but the specified library is installed.
Most C libraries used with `distutils` are only used to build python
extensions, but libraries built through this method will be installed
so that they can be reused by third-party packages.
Parameters
----------
name : str
Name of the installed library.
sources : sequence
List of the library's source files. See `add_library` for details.
install_dir : str
Path to install the library, relative to the current sub-package.
build_info : dict, optional
The following keys are allowed:
* depends
* macros
* include_dirs
* extra_compiler_args
* extra_f77_compile_args
* extra_f90_compile_args
* f2py_options
* language
Returns
-------
None
See Also
--------
add_library, add_npy_pkg_config, get_info
Notes
-----
The best way to encode the options required to link against the specified
C libraries is to use a "libname.ini" file, and use `get_info` to
retrieve the required options (see `add_npy_pkg_config` for more
information).
"""
if not build_info:
build_info = {}
install_dir = os.path.join(self.package_path, install_dir)
self._add_library(name, sources, install_dir, build_info)
self.installed_libraries.append(InstallableLib(name, build_info, install_dir))
def add_npy_pkg_config(self, template, install_dir, subst_dict=None):
"""
Generate and install a npy-pkg config file from a template.
The config file generated from `template` is installed in the
given install directory, using `subst_dict` for variable substitution.
Parameters
----------
template : str
The path of the template, relatively to the current package path.
install_dir : str
Where to install the npy-pkg config file, relatively to the current
package path.
subst_dict : dict, optional
If given, any string of the form ``@key@`` will be replaced by
``subst_dict[key]`` in the template file when installed. The install
prefix is always available through the variable ``@prefix@``, since the
install prefix is not easy to get reliably from setup.py.
See also
--------
add_installed_library, get_info
Notes
-----
This works for both standard installs and in-place builds, i.e. the
``@prefix@`` refer to the source directory for in-place builds.
Examples
--------
::
config.add_npy_pkg_config('foo.ini.in', 'lib', {'foo': bar})
Assuming the foo.ini.in file has the following content::
[meta]
Name=@foo@
Version=1.0
Description=dummy description
[default]
Cflags=-I@prefix@/include
Libs=
The generated file will have the following content::
[meta]
Name=bar
Version=1.0
Description=dummy description
[default]
Cflags=-Iprefix_dir/include
Libs=
and will be installed as foo.ini in the 'lib' subpath.
When cross-compiling with numpy distutils, it might be necessary to
use modified npy-pkg-config files. Using the default/generated files
will link with the host libraries (i.e. libnpymath.a). For
cross-compilation you of-course need to link with target libraries,
while using the host Python installation.
You can copy out the numpy/core/lib/npy-pkg-config directory, add a
pkgdir value to the .ini files and set NPY_PKG_CONFIG_PATH environment
variable to point to the directory with the modified npy-pkg-config
files.
Example npymath.ini modified for cross-compilation::
[meta]
Name=npymath
Description=Portable, core math library implementing C99 standard
Version=0.1
[variables]
pkgname=numpy.core
pkgdir=/build/arm-linux-gnueabi/sysroot/usr/lib/python3.7/site-packages/numpy/core
prefix=${pkgdir}
libdir=${prefix}/lib
includedir=${prefix}/include
[default]
Libs=-L${libdir} -lnpymath
Cflags=-I${includedir}
Requires=mlib
[msvc]
Libs=/LIBPATH:${libdir} npymath.lib
Cflags=/INCLUDE:${includedir}
Requires=mlib
"""
if subst_dict is None:
subst_dict = {}
template = os.path.join(self.package_path, template)
if self.name in self.installed_pkg_config:
self.installed_pkg_config[self.name].append((template, install_dir,
subst_dict))
else:
self.installed_pkg_config[self.name] = [(template, install_dir,
subst_dict)]
def add_scripts(self,*files):
"""Add scripts to configuration.
Add the sequence of files to the beginning of the scripts list.
Scripts will be installed under the <prefix>/bin/ directory.
"""
scripts = self.paths(files)
dist = self.get_distribution()
if dist is not None:
if dist.scripts is None:
dist.scripts = []
dist.scripts.extend(scripts)
else:
self.scripts.extend(scripts)
def dict_append(self,**dict):
for key in self.list_keys:
a = getattr(self, key)
a.extend(dict.get(key, []))
for key in self.dict_keys:
a = getattr(self, key)
a.update(dict.get(key, {}))
known_keys = self.list_keys + self.dict_keys + self.extra_keys
for key in dict.keys():
if key not in known_keys:
a = getattr(self, key, None)
if a and a==dict[key]: continue
self.warn('Inheriting attribute %r=%r from %r' \
% (key, dict[key], dict.get('name', '?')))
setattr(self, key, dict[key])
self.extra_keys.append(key)
elif key in self.extra_keys:
self.info('Ignoring attempt to set %r (from %r to %r)' \
% (key, getattr(self, key), dict[key]))
elif key in known_keys:
# key is already processed above
pass
else:
raise ValueError("Don't know about key=%r" % (key))
def __str__(self):
from pprint import pformat
known_keys = self.list_keys + self.dict_keys + self.extra_keys
s = '<'+5*'-' + '\n'
s += 'Configuration of '+self.name+':\n'
known_keys.sort()
for k in known_keys:
a = getattr(self, k, None)
if a:
s += '%s = %s\n' % (k, pformat(a))
s += 5*'-' + '>'
return s
def get_config_cmd(self):
"""
Returns the numpy.distutils config command instance.
"""
cmd = get_cmd('config')
cmd.ensure_finalized()
cmd.dump_source = 0
cmd.noisy = 0
old_path = os.environ.get('PATH')
if old_path:
path = os.pathsep.join(['.', old_path])
os.environ['PATH'] = path
return cmd
def get_build_temp_dir(self):
"""
Return a path to a temporary directory where temporary files should be
placed.
"""
cmd = get_cmd('build')
cmd.ensure_finalized()
return cmd.build_temp
def have_f77c(self):
"""Check for availability of Fortran 77 compiler.
Use it inside source generating function to ensure that
setup distribution instance has been initialized.
Notes
-----
True if a Fortran 77 compiler is available (because a simple Fortran 77
code was able to be compiled successfully).
"""
simple_fortran_subroutine = '''
subroutine simple
end
'''
config_cmd = self.get_config_cmd()
flag = config_cmd.try_compile(simple_fortran_subroutine, lang='f77')
return flag
def have_f90c(self):
"""Check for availability of Fortran 90 compiler.
Use it inside source generating function to ensure that
setup distribution instance has been initialized.
Notes
-----
True if a Fortran 90 compiler is available (because a simple Fortran
90 code was able to be compiled successfully)
"""
simple_fortran_subroutine = '''
subroutine simple
end
'''
config_cmd = self.get_config_cmd()
flag = config_cmd.try_compile(simple_fortran_subroutine, lang='f90')
return flag
def append_to(self, extlib):
"""Append libraries, include_dirs to extension or library item.
"""
if is_sequence(extlib):
lib_name, build_info = extlib
dict_append(build_info,
libraries=self.libraries,
include_dirs=self.include_dirs)
else:
from numpy.distutils.core import Extension
assert isinstance(extlib, Extension), repr(extlib)
extlib.libraries.extend(self.libraries)
extlib.include_dirs.extend(self.include_dirs)
def _get_svn_revision(self, path):
"""Return path's SVN revision number.
"""
try:
output = subprocess.check_output(['svnversion'], cwd=path)
except (subprocess.CalledProcessError, OSError):
pass
else:
m = re.match(rb'(?P<revision>\d+)', output)
if m:
return int(m.group('revision'))
if sys.platform=='win32' and os.environ.get('SVN_ASP_DOT_NET_HACK', None):
entries = njoin(path, '_svn', 'entries')
else:
entries = njoin(path, '.svn', 'entries')
if os.path.isfile(entries):
with open(entries) as f:
fstr = f.read()
if fstr[:5] == '<?xml': # pre 1.4
m = re.search(r'revision="(?P<revision>\d+)"', fstr)
if m:
return int(m.group('revision'))
else: # non-xml entries file --- check to be sure that
m = re.search(r'dir[\n\r]+(?P<revision>\d+)', fstr)
if m:
return int(m.group('revision'))
return None
def _get_hg_revision(self, path):
"""Return path's Mercurial revision number.
"""
try:
output = subprocess.check_output(
['hg', 'identify', '--num'], cwd=path)
except (subprocess.CalledProcessError, OSError):
pass
else:
m = re.match(rb'(?P<revision>\d+)', output)
if m:
return int(m.group('revision'))
branch_fn = njoin(path, '.hg', 'branch')
branch_cache_fn = njoin(path, '.hg', 'branch.cache')
if os.path.isfile(branch_fn):
branch0 = None
with open(branch_fn) as f:
revision0 = f.read().strip()
branch_map = {}
with open(branch_cache_fn, 'r') as f:
for line in f:
branch1, revision1 = line.split()[:2]
if revision1==revision0:
branch0 = branch1
try:
revision1 = int(revision1)
except ValueError:
continue
branch_map[branch1] = revision1
return branch_map.get(branch0)
return None
def get_version(self, version_file=None, version_variable=None):
"""Try to get version string of a package.
Return a version string of the current package or None if the version
information could not be detected.
Notes
-----
This method scans files named
__version__.py, <packagename>_version.py, version.py, and
__svn_version__.py for string variables version, __version__, and
<packagename>_version, until a version number is found.
"""
version = getattr(self, 'version', None)
if version is not None:
return version
# Get version from version file.
if version_file is None:
files = ['__version__.py',
self.name.split('.')[-1]+'_version.py',
'version.py',
'__svn_version__.py',
'__hg_version__.py']
else:
files = [version_file]
if version_variable is None:
version_vars = ['version',
'__version__',
self.name.split('.')[-1]+'_version']
else:
version_vars = [version_variable]
for f in files:
fn = njoin(self.local_path, f)
if os.path.isfile(fn):
info = ('.py', 'U', 1)
name = os.path.splitext(os.path.basename(fn))[0]
n = dot_join(self.name, name)
try:
version_module = exec_mod_from_location(
'_'.join(n.split('.')), fn)
except ImportError as e:
self.warn(str(e))
version_module = None
if version_module is None:
continue
for a in version_vars:
version = getattr(version_module, a, None)
if version is not None:
break
# Try if versioneer module
try:
version = version_module.get_versions()['version']
except AttributeError:
pass
if version is not None:
break
if version is not None:
self.version = version
return version
# Get version as SVN or Mercurial revision number
revision = self._get_svn_revision(self.local_path)
if revision is None:
revision = self._get_hg_revision(self.local_path)
if revision is not None:
version = str(revision)
self.version = version
return version
def make_svn_version_py(self, delete=True):
"""Appends a data function to the data_files list that will generate
__svn_version__.py file to the current package directory.
Generate package __svn_version__.py file from SVN revision number,
it will be removed after python exits but will be available
when sdist, etc commands are executed.
Notes
-----
If __svn_version__.py existed before, nothing is done.
This is
intended for working with source directories that are in an SVN
repository.
"""
target = njoin(self.local_path, '__svn_version__.py')
revision = self._get_svn_revision(self.local_path)
if os.path.isfile(target) or revision is None:
return
else:
def generate_svn_version_py():
if not os.path.isfile(target):
version = str(revision)
self.info('Creating %s (version=%r)' % (target, version))
with open(target, 'w') as f:
f.write('version = %r\n' % (version))
def rm_file(f=target,p=self.info):
if delete:
try: os.remove(f); p('removed '+f)
except OSError: pass
try: os.remove(f+'c'); p('removed '+f+'c')
except OSError: pass
atexit.register(rm_file)
return target
self.add_data_files(('', generate_svn_version_py()))
def make_hg_version_py(self, delete=True):
"""Appends a data function to the data_files list that will generate
__hg_version__.py file to the current package directory.
Generate package __hg_version__.py file from Mercurial revision,
it will be removed after python exits but will be available
when sdist, etc commands are executed.
Notes
-----
If __hg_version__.py existed before, nothing is done.
This is intended for working with source directories that are
in an Mercurial repository.
"""
target = njoin(self.local_path, '__hg_version__.py')
revision = self._get_hg_revision(self.local_path)
if os.path.isfile(target) or revision is None:
return
else:
def generate_hg_version_py():
if not os.path.isfile(target):
version = str(revision)
self.info('Creating %s (version=%r)' % (target, version))
with open(target, 'w') as f:
f.write('version = %r\n' % (version))
def rm_file(f=target,p=self.info):
if delete:
try: os.remove(f); p('removed '+f)
except OSError: pass
try: os.remove(f+'c'); p('removed '+f+'c')
except OSError: pass
atexit.register(rm_file)
return target
self.add_data_files(('', generate_hg_version_py()))
def make_config_py(self,name='__config__'):
"""Generate package __config__.py file containing system_info
information used during building the package.
This file is installed to the
package installation directory.
"""
self.py_modules.append((self.name, name, generate_config_py))
def get_info(self,*names):
"""Get resources information.
Return information (from system_info.get_info) for all of the names in
the argument list in a single dictionary.
"""
from .system_info import get_info, dict_append
info_dict = {}
for a in names:
dict_append(info_dict,**get_info(a))
return info_dict
def configuration(parent_package='',top_path=None):
from numpy.distutils.misc_util import Configuration
config = Configuration('polynomial', parent_package, top_path)
config.add_subpackage('tests')
config.add_data_files('*.pyi')
return config | null |
169,742 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermadd(c1, c2):
"""
Add one Hermite series to another.
Returns the sum of two Hermite series `c1` + `c2`. The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the Hermite series of their sum.
See Also
--------
hermsub, hermmulx, hermmul, hermdiv, hermpow
Notes
-----
Unlike multiplication, division, etc., the sum of two Hermite series
is a Hermite series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."
Examples
--------
>>> from numpy.polynomial.hermite import hermadd
>>> hermadd([1, 2, 3], [1, 2, 3, 4])
array([2., 4., 6., 4.])
"""
return pu._add(c1, c2)
def hermmulx(c):
"""Multiply a Hermite series by x.
Multiply the Hermite series `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
See Also
--------
hermadd, hermsub, hermmul, hermdiv, hermpow
Notes
-----
The multiplication uses the recursion relationship for Hermite
polynomials in the form
.. math::
xP_i(x) = (P_{i + 1}(x)/2 + i*P_{i - 1}(x))
Examples
--------
>>> from numpy.polynomial.hermite import hermmulx
>>> hermmulx([1, 2, 3])
array([2. , 6.5, 1. , 1.5])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]*0
prd[1] = c[0]/2
for i in range(1, len(c)):
prd[i + 1] = c[i]/2
prd[i - 1] += c[i]*i
return prd
The provided code snippet includes necessary dependencies for implementing the `poly2herm` function. Write a Python function `def poly2herm(pol)` to solve the following problem:
poly2herm(pol) Convert a polynomial to a Hermite series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Hermite series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Hermite series. See Also -------- herm2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.hermite import poly2herm >>> poly2herm(np.arange(4)) array([1. , 2.75 , 0.5 , 0.375])
Here is the function:
def poly2herm(pol):
"""
poly2herm(pol)
Convert a polynomial to a Hermite series.
Convert an array representing the coefficients of a polynomial (relative
to the "standard" basis) ordered from lowest degree to highest, to an
array of the coefficients of the equivalent Hermite series, ordered
from lowest to highest degree.
Parameters
----------
pol : array_like
1-D array containing the polynomial coefficients
Returns
-------
c : ndarray
1-D array containing the coefficients of the equivalent Hermite
series.
See Also
--------
herm2poly
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy.polynomial.hermite import poly2herm
>>> poly2herm(np.arange(4))
array([1. , 2.75 , 0.5 , 0.375])
"""
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
for i in range(deg, -1, -1):
res = hermadd(hermmulx(res), pol[i])
return res | poly2herm(pol) Convert a polynomial to a Hermite series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Hermite series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Hermite series. See Also -------- herm2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.hermite import poly2herm >>> poly2herm(np.arange(4)) array([1. , 2.75 , 0.5 , 0.375]) |
169,743 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyadd(c1, c2):
"""
Add one polynomial to another.
Returns the sum of two polynomials `c1` + `c2`. The arguments are
sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to high.
Returns
-------
out : ndarray
The coefficient array representing their sum.
See Also
--------
polysub, polymulx, polymul, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> sum = P.polyadd(c1,c2); sum
array([4., 4., 4.])
>>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2)
28.0
"""
return pu._add(c1, c2)
def polysub(c1, c2):
"""
Subtract one polynomial from another.
Returns the difference of two polynomials `c1` - `c2`. The arguments
are sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of coefficients representing their difference.
See Also
--------
polyadd, polymulx, polymul, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polysub(c1,c2)
array([-2., 0., 2.])
>>> P.polysub(c2,c1) # -P.polysub(c1,c2)
array([ 2., 0., -2.])
"""
return pu._sub(c1, c2)
def polymulx(c):
"""Multiply a polynomial by x.
Multiply the polynomial `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
See Also
--------
polyadd, polysub, polymul, polydiv, polypow
Notes
-----
.. versionadded:: 1.5.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]*0
prd[1:] = c
return prd
The provided code snippet includes necessary dependencies for implementing the `herm2poly` function. Write a Python function `def herm2poly(c)` to solve the following problem:
Convert a Hermite series to a polynomial. Convert an array representing the coefficients of a Hermite series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- c : array_like 1-D array containing the Hermite series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2herm Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.hermite import herm2poly >>> herm2poly([ 1. , 2.75 , 0.5 , 0.375]) array([0., 1., 2., 3.])
Here is the function:
def herm2poly(c):
"""
Convert a Hermite series to a polynomial.
Convert an array representing the coefficients of a Hermite series,
ordered from lowest degree to highest, to an array of the coefficients
of the equivalent polynomial (relative to the "standard" basis) ordered
from lowest to highest degree.
Parameters
----------
c : array_like
1-D array containing the Hermite series coefficients, ordered
from lowest order term to highest.
Returns
-------
pol : ndarray
1-D array containing the coefficients of the equivalent polynomial
(relative to the "standard" basis) ordered from lowest order term
to highest.
See Also
--------
poly2herm
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy.polynomial.hermite import herm2poly
>>> herm2poly([ 1. , 2.75 , 0.5 , 0.375])
array([0., 1., 2., 3.])
"""
from .polynomial import polyadd, polysub, polymulx
[c] = pu.as_series([c])
n = len(c)
if n == 1:
return c
if n == 2:
c[1] *= 2
return c
else:
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], c1*(2*(i - 1)))
c1 = polyadd(tmp, polymulx(c1)*2)
return polyadd(c0, polymulx(c1)*2) | Convert a Hermite series to a polynomial. Convert an array representing the coefficients of a Hermite series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- c : array_like 1-D array containing the Hermite series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2herm Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.hermite import herm2poly >>> herm2poly([ 1. , 2.75 , 0.5 , 0.375]) array([0., 1., 2., 3.]) |
169,744 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermline(off, scl):
"""
Hermite series whose graph is a straight line.
Parameters
----------
off, scl : scalars
The specified line is given by ``off + scl*x``.
Returns
-------
y : ndarray
This module's representation of the Hermite series for
``off + scl*x``.
See Also
--------
numpy.polynomial.polynomial.polyline
numpy.polynomial.chebyshev.chebline
numpy.polynomial.legendre.legline
numpy.polynomial.laguerre.lagline
numpy.polynomial.hermite_e.hermeline
Examples
--------
>>> from numpy.polynomial.hermite import hermline, hermval
>>> hermval(0,hermline(3, 2))
3.0
>>> hermval(1,hermline(3, 2))
5.0
"""
if scl != 0:
return np.array([off, scl/2])
else:
return np.array([off])
def hermmul(c1, c2):
"""
Multiply one Hermite series by another.
Returns the product of two Hermite series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Hermite series coefficients representing their product.
See Also
--------
hermadd, hermsub, hermmulx, hermdiv, hermpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Hermite polynomial basis set. Thus, to express
the product as a Hermite series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial.hermite import hermmul
>>> hermmul([1, 2, 3], [0, 1, 2])
array([52., 29., 52., 7., 6.])
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = hermsub(c[-i]*xs, c1*(2*(nd - 1)))
c1 = hermadd(tmp, hermmulx(c1)*2)
return hermadd(c0, hermmulx(c1)*2)
The provided code snippet includes necessary dependencies for implementing the `hermfromroots` function. Write a Python function `def hermfromroots(roots)` to solve the following problem:
Generate a Hermite series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in Hermite form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x) The coefficient of the last term is not generally 1 for monic polynomials in Hermite form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below). See Also -------- numpy.polynomial.polynomial.polyfromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.laguerre.lagfromroots numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.hermite_e.hermefromroots Examples -------- >>> from numpy.polynomial.hermite import hermfromroots, hermval >>> coef = hermfromroots((-1, 0, 1)) >>> hermval((-1, 0, 1), coef) array([0., 0., 0.]) >>> coef = hermfromroots((-1j, 1j)) >>> hermval((-1j, 1j), coef) array([0.+0.j, 0.+0.j])
Here is the function:
def hermfromroots(roots):
"""
Generate a Hermite series with given roots.
The function returns the coefficients of the polynomial
.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
in Hermite form, where the `r_n` are the roots specified in `roots`.
If a zero has multiplicity n, then it must appear in `roots` n times.
For instance, if 2 is a root of multiplicity three and 3 is a root of
multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
roots can appear in any order.
If the returned coefficients are `c`, then
.. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x)
The coefficient of the last term is not generally 1 for monic
polynomials in Hermite form.
Parameters
----------
roots : array_like
Sequence containing the roots.
Returns
-------
out : ndarray
1-D array of coefficients. If all roots are real then `out` is a
real array, if some of the roots are complex, then `out` is complex
even if all the coefficients in the result are real (see Examples
below).
See Also
--------
numpy.polynomial.polynomial.polyfromroots
numpy.polynomial.legendre.legfromroots
numpy.polynomial.laguerre.lagfromroots
numpy.polynomial.chebyshev.chebfromroots
numpy.polynomial.hermite_e.hermefromroots
Examples
--------
>>> from numpy.polynomial.hermite import hermfromroots, hermval
>>> coef = hermfromroots((-1, 0, 1))
>>> hermval((-1, 0, 1), coef)
array([0., 0., 0.])
>>> coef = hermfromroots((-1j, 1j))
>>> hermval((-1j, 1j), coef)
array([0.+0.j, 0.+0.j])
"""
return pu._fromroots(hermline, hermmul, roots) | Generate a Hermite series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in Hermite form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x) The coefficient of the last term is not generally 1 for monic polynomials in Hermite form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below). See Also -------- numpy.polynomial.polynomial.polyfromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.laguerre.lagfromroots numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.hermite_e.hermefromroots Examples -------- >>> from numpy.polynomial.hermite import hermfromroots, hermval >>> coef = hermfromroots((-1, 0, 1)) >>> hermval((-1, 0, 1), coef) array([0., 0., 0.]) >>> coef = hermfromroots((-1j, 1j)) >>> hermval((-1j, 1j), coef) array([0.+0.j, 0.+0.j]) |
169,745 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermmul(c1, c2):
"""
Multiply one Hermite series by another.
Returns the product of two Hermite series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Hermite series coefficients representing their product.
See Also
--------
hermadd, hermsub, hermmulx, hermdiv, hermpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Hermite polynomial basis set. Thus, to express
the product as a Hermite series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial.hermite import hermmul
>>> hermmul([1, 2, 3], [0, 1, 2])
array([52., 29., 52., 7., 6.])
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = hermsub(c[-i]*xs, c1*(2*(nd - 1)))
c1 = hermadd(tmp, hermmulx(c1)*2)
return hermadd(c0, hermmulx(c1)*2)
The provided code snippet includes necessary dependencies for implementing the `hermdiv` function. Write a Python function `def hermdiv(c1, c2)` to solve the following problem:
Divide one Hermite series by another. Returns the quotient-with-remainder of two Hermite series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Hermite series coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of Hermite series coefficients representing the quotient and remainder. See Also -------- hermadd, hermsub, hermmulx, hermmul, hermpow Notes ----- In general, the (polynomial) division of one Hermite series by another results in quotient and remainder terms that are not in the Hermite polynomial basis set. Thus, to express these results as a Hermite series, it is necessary to "reproject" the results onto the Hermite basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite import hermdiv >>> hermdiv([ 52., 29., 52., 7., 6.], [0, 1, 2]) (array([1., 2., 3.]), array([0.])) >>> hermdiv([ 54., 31., 52., 7., 6.], [0, 1, 2]) (array([1., 2., 3.]), array([2., 2.])) >>> hermdiv([ 53., 30., 52., 7., 6.], [0, 1, 2]) (array([1., 2., 3.]), array([1., 1.]))
Here is the function:
def hermdiv(c1, c2):
"""
Divide one Hermite series by another.
Returns the quotient-with-remainder of two Hermite series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarrays
Of Hermite series coefficients representing the quotient and
remainder.
See Also
--------
hermadd, hermsub, hermmulx, hermmul, hermpow
Notes
-----
In general, the (polynomial) division of one Hermite series by another
results in quotient and remainder terms that are not in the Hermite
polynomial basis set. Thus, to express these results as a Hermite
series, it is necessary to "reproject" the results onto the Hermite
basis set, which may produce "unintuitive" (but correct) results; see
Examples section below.
Examples
--------
>>> from numpy.polynomial.hermite import hermdiv
>>> hermdiv([ 52., 29., 52., 7., 6.], [0, 1, 2])
(array([1., 2., 3.]), array([0.]))
>>> hermdiv([ 54., 31., 52., 7., 6.], [0, 1, 2])
(array([1., 2., 3.]), array([2., 2.]))
>>> hermdiv([ 53., 30., 52., 7., 6.], [0, 1, 2])
(array([1., 2., 3.]), array([1., 1.]))
"""
return pu._div(hermmul, c1, c2) | Divide one Hermite series by another. Returns the quotient-with-remainder of two Hermite series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Hermite series coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of Hermite series coefficients representing the quotient and remainder. See Also -------- hermadd, hermsub, hermmulx, hermmul, hermpow Notes ----- In general, the (polynomial) division of one Hermite series by another results in quotient and remainder terms that are not in the Hermite polynomial basis set. Thus, to express these results as a Hermite series, it is necessary to "reproject" the results onto the Hermite basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite import hermdiv >>> hermdiv([ 52., 29., 52., 7., 6.], [0, 1, 2]) (array([1., 2., 3.]), array([0.])) >>> hermdiv([ 54., 31., 52., 7., 6.], [0, 1, 2]) (array([1., 2., 3.]), array([2., 2.])) >>> hermdiv([ 53., 30., 52., 7., 6.], [0, 1, 2]) (array([1., 2., 3.]), array([1., 1.])) |
169,746 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermmul(c1, c2):
"""
Multiply one Hermite series by another.
Returns the product of two Hermite series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Hermite series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Hermite series coefficients representing their product.
See Also
--------
hermadd, hermsub, hermmulx, hermdiv, hermpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Hermite polynomial basis set. Thus, to express
the product as a Hermite series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial.hermite import hermmul
>>> hermmul([1, 2, 3], [0, 1, 2])
array([52., 29., 52., 7., 6.])
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = hermsub(c[-i]*xs, c1*(2*(nd - 1)))
c1 = hermadd(tmp, hermmulx(c1)*2)
return hermadd(c0, hermmulx(c1)*2)
The provided code snippet includes necessary dependencies for implementing the `hermpow` function. Write a Python function `def hermpow(c, pow, maxpower=16)` to solve the following problem:
Raise a Hermite series to a power. Returns the Hermite series `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` Parameters ---------- c : array_like 1-D array of Hermite series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Hermite series of power. See Also -------- hermadd, hermsub, hermmulx, hermmul, hermdiv Examples -------- >>> from numpy.polynomial.hermite import hermpow >>> hermpow([1, 2, 3], 2) array([81., 52., 82., 12., 9.])
Here is the function:
def hermpow(c, pow, maxpower=16):
"""Raise a Hermite series to a power.
Returns the Hermite series `c` raised to the power `pow`. The
argument `c` is a sequence of coefficients ordered from low to high.
i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.``
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to
high.
pow : integer
Power to which the series will be raised
maxpower : integer, optional
Maximum power allowed. This is mainly to limit growth of the series
to unmanageable size. Default is 16
Returns
-------
coef : ndarray
Hermite series of power.
See Also
--------
hermadd, hermsub, hermmulx, hermmul, hermdiv
Examples
--------
>>> from numpy.polynomial.hermite import hermpow
>>> hermpow([1, 2, 3], 2)
array([81., 52., 82., 12., 9.])
"""
return pu._pow(hermmul, c, pow, maxpower) | Raise a Hermite series to a power. Returns the Hermite series `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` Parameters ---------- c : array_like 1-D array of Hermite series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Hermite series of power. See Also -------- hermadd, hermsub, hermmulx, hermmul, hermdiv Examples -------- >>> from numpy.polynomial.hermite import hermpow >>> hermpow([1, 2, 3], 2) array([81., 52., 82., 12., 9.]) |
169,747 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `hermder` function. Write a Python function `def hermder(c, m=1, scl=1, axis=0)` to solve the following problem:
Differentiate a Hermite series. Returns the Hermite series coefficients `c` differentiated `m` times along `axis`. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``1*H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Hermite series coefficients. If `c` is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by `scl`. The end result is multiplication by ``scl**m``. This is for use in a linear change of variable. (Default: 1) axis : int, optional Axis over which the derivative is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- der : ndarray Hermite series of the derivative. See Also -------- hermint Notes ----- In general, the result of differentiating a Hermite series does not resemble the same operation on a power series. Thus the result of this function may be "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite import hermder >>> hermder([ 1. , 0.5, 0.5, 0.5]) array([1., 2., 3.]) >>> hermder([-0.5, 1./2., 1./8., 1./12., 1./16.], m=2) array([1., 2., 3.])
Here is the function:
def hermder(c, m=1, scl=1, axis=0):
"""
Differentiate a Hermite series.
Returns the Hermite series coefficients `c` differentiated `m` times
along `axis`. At each iteration the result is multiplied by `scl` (the
scaling factor is for use in a linear change of variable). The argument
`c` is an array of coefficients from low to high degree along each
axis, e.g., [1,2,3] represents the series ``1*H_0 + 2*H_1 + 3*H_2``
while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) +
2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is
``y``.
Parameters
----------
c : array_like
Array of Hermite series coefficients. If `c` is multidimensional the
different axis correspond to different variables with the degree in
each axis given by the corresponding index.
m : int, optional
Number of derivatives taken, must be non-negative. (Default: 1)
scl : scalar, optional
Each differentiation is multiplied by `scl`. The end result is
multiplication by ``scl**m``. This is for use in a linear change of
variable. (Default: 1)
axis : int, optional
Axis over which the derivative is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
der : ndarray
Hermite series of the derivative.
See Also
--------
hermint
Notes
-----
In general, the result of differentiating a Hermite series does not
resemble the same operation on a power series. Thus the result of this
function may be "unintuitive," albeit correct; see Examples section
below.
Examples
--------
>>> from numpy.polynomial.hermite import hermder
>>> hermder([ 1. , 0.5, 0.5, 0.5])
array([1., 2., 3.])
>>> hermder([-0.5, 1./2., 1./8., 1./12., 1./16.], m=2)
array([1., 2., 3.])
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
cnt = pu._deprecate_as_int(m, "the order of derivation")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of derivation must be non-negative")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
n = len(c)
if cnt >= n:
c = c[:1]*0
else:
for i in range(cnt):
n = n - 1
c *= scl
der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
for j in range(n, 0, -1):
der[j - 1] = (2*j)*c[j]
c = der
c = np.moveaxis(c, 0, iaxis)
return c | Differentiate a Hermite series. Returns the Hermite series coefficients `c` differentiated `m` times along `axis`. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``1*H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Hermite series coefficients. If `c` is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by `scl`. The end result is multiplication by ``scl**m``. This is for use in a linear change of variable. (Default: 1) axis : int, optional Axis over which the derivative is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- der : ndarray Hermite series of the derivative. See Also -------- hermint Notes ----- In general, the result of differentiating a Hermite series does not resemble the same operation on a power series. Thus the result of this function may be "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite import hermder >>> hermder([ 1. , 0.5, 0.5, 0.5]) array([1., 2., 3.]) >>> hermder([-0.5, 1./2., 1./8., 1./12., 1./16.], m=2) array([1., 2., 3.]) |
169,748 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermval(x, c, tensor=True):
"""
Evaluate an Hermite series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
hermval2d, hermgrid2d, hermval3d, hermgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.hermite import hermval
>>> coef = [1,2,3]
>>> hermval(1, coef)
11.0
>>> hermval([[1,2],[3,4]], coef)
array([[ 11., 51.],
[115., 203.]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
x2 = x*2
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - c1*(2*(nd - 1))
c1 = tmp + c1*x2
return c0 + c1*x2
The provided code snippet includes necessary dependencies for implementing the `hermint` function. Write a Python function `def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0)` to solve the following problem:
Integrate a Hermite series. Returns the Hermite series coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Hermite series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at ``lbnd`` is the first value in the list, the value of the second integral at ``lbnd`` is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray Hermite series coefficients of the integral. Raises ------ ValueError If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- hermder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a` - perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite import hermint >>> hermint([1,2,3]) # integrate once, value 0 at 0. array([1. , 0.5, 0.5, 0.5]) >>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0 array([-0.5 , 0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary >>> hermint([1,2,3], k=1) # integrate once, value 1 at 0. array([2. , 0.5, 0.5, 0.5]) >>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1 array([-2. , 0.5, 0.5, 0.5]) >>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1) array([ 1.66666667, -0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary
Here is the function:
def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
"""
Integrate a Hermite series.
Returns the Hermite series coefficients `c` integrated `m` times from
`lbnd` along `axis`. At each iteration the resulting series is
**multiplied** by `scl` and an integration constant, `k`, is added.
The scaling factor is for use in a linear change of variable. ("Buyer
beware": note that, depending on what one is doing, one may want `scl`
to be the reciprocal of what one might expect; for more information,
see the Notes section below.) The argument `c` is an array of
coefficients from low to high degree along each axis, e.g., [1,2,3]
represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]]
represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) +
2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
Parameters
----------
c : array_like
Array of Hermite series coefficients. If c is multidimensional the
different axis correspond to different variables with the degree in
each axis given by the corresponding index.
m : int, optional
Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
Integration constant(s). The value of the first integral at
``lbnd`` is the first value in the list, the value of the second
integral at ``lbnd`` is the second value, etc. If ``k == []`` (the
default), all constants are set to zero. If ``m == 1``, a single
scalar can be given instead of a list.
lbnd : scalar, optional
The lower bound of the integral. (Default: 0)
scl : scalar, optional
Following each integration the result is *multiplied* by `scl`
before the integration constant is added. (Default: 1)
axis : int, optional
Axis over which the integral is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
S : ndarray
Hermite series coefficients of the integral.
Raises
------
ValueError
If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
``np.ndim(scl) != 0``.
See Also
--------
hermder
Notes
-----
Note that the result of each integration is *multiplied* by `scl`.
Why is this important to note? Say one is making a linear change of
variable :math:`u = ax + b` in an integral relative to `x`. Then
:math:`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a` - perhaps not what one would have first thought.
Also note that, in general, the result of integrating a C-series needs
to be "reprojected" onto the C-series basis set. Thus, typically,
the result of this function is "unintuitive," albeit correct; see
Examples section below.
Examples
--------
>>> from numpy.polynomial.hermite import hermint
>>> hermint([1,2,3]) # integrate once, value 0 at 0.
array([1. , 0.5, 0.5, 0.5])
>>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0
array([-0.5 , 0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary
>>> hermint([1,2,3], k=1) # integrate once, value 1 at 0.
array([2. , 0.5, 0.5, 0.5])
>>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1
array([-2. , 0.5, 0.5, 0.5])
>>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1)
array([ 1.66666667, -0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if not np.iterable(k):
k = [k]
cnt = pu._deprecate_as_int(m, "the order of integration")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of integration must be non-negative")
if len(k) > cnt:
raise ValueError("Too many integration constants")
if np.ndim(lbnd) != 0:
raise ValueError("lbnd must be a scalar.")
if np.ndim(scl) != 0:
raise ValueError("scl must be a scalar.")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
k = list(k) + [0]*(cnt - len(k))
for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
c[0] += k[i]
else:
tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
tmp[0] = c[0]*0
tmp[1] = c[0]/2
for j in range(1, n):
tmp[j + 1] = c[j]/(2*(j + 1))
tmp[0] += k[i] - hermval(lbnd, tmp)
c = tmp
c = np.moveaxis(c, 0, iaxis)
return c | Integrate a Hermite series. Returns the Hermite series coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Hermite series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at ``lbnd`` is the first value in the list, the value of the second integral at ``lbnd`` is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray Hermite series coefficients of the integral. Raises ------ ValueError If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- hermder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a` - perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial.hermite import hermint >>> hermint([1,2,3]) # integrate once, value 0 at 0. array([1. , 0.5, 0.5, 0.5]) >>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0 array([-0.5 , 0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary >>> hermint([1,2,3], k=1) # integrate once, value 1 at 0. array([2. , 0.5, 0.5, 0.5]) >>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1 array([-2. , 0.5, 0.5, 0.5]) >>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1) array([ 1.66666667, -0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary |
169,749 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermval(x, c, tensor=True):
"""
Evaluate an Hermite series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
hermval2d, hermgrid2d, hermval3d, hermgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.hermite import hermval
>>> coef = [1,2,3]
>>> hermval(1, coef)
11.0
>>> hermval([[1,2],[3,4]], coef)
array([[ 11., 51.],
[115., 203.]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
x2 = x*2
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - c1*(2*(nd - 1))
c1 = tmp + c1*x2
return c0 + c1*x2
The provided code snippet includes necessary dependencies for implementing the `hermval2d` function. Write a Python function `def hermval2d(x, y, c)` to solve the following problem:
Evaluate a 2-D Hermite series at points (x, y). This function returns the values: .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_j(y) The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points formed with pairs of corresponding values from `x` and `y`. See Also -------- hermval, hermgrid2d, hermval3d, hermgrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def hermval2d(x, y, c):
"""
Evaluate a 2-D Hermite series at points (x, y).
This function returns the values:
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_j(y)
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars and they
must have the same shape after conversion. In either case, either `x`
and `y` or their elements must support multiplication and addition both
with themselves and with the elements of `c`.
If `c` is a 1-D array a one is implicitly appended to its shape to make
it 2-D. The shape of the result will be c.shape[2:] + x.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points `(x, y)`,
where `x` and `y` must have the same shape. If `x` or `y` is a list
or tuple, it is first converted to an ndarray, otherwise it is left
unchanged and if it isn't an ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term
of multi-degree i,j is contained in ``c[i,j]``. If `c` has
dimension greater than two the remaining indices enumerate multiple
sets of coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points formed with
pairs of corresponding values from `x` and `y`.
See Also
--------
hermval, hermgrid2d, hermval3d, hermgrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(hermval, c, x, y) | Evaluate a 2-D Hermite series at points (x, y). This function returns the values: .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_j(y) The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points formed with pairs of corresponding values from `x` and `y`. See Also -------- hermval, hermgrid2d, hermval3d, hermgrid3d Notes ----- .. versionadded:: 1.7.0 |
169,750 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermval(x, c, tensor=True):
"""
Evaluate an Hermite series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
hermval2d, hermgrid2d, hermval3d, hermgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.hermite import hermval
>>> coef = [1,2,3]
>>> hermval(1, coef)
11.0
>>> hermval([[1,2],[3,4]], coef)
array([[ 11., 51.],
[115., 203.]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
x2 = x*2
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - c1*(2*(nd - 1))
c1 = tmp + c1*x2
return c0 + c1*x2
The provided code snippet includes necessary dependencies for implementing the `hermgrid2d` function. Write a Python function `def hermgrid2d(x, y, c)` to solve the following problem:
Evaluate a 2-D Hermite series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_j(b) where the points `(a, b)` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- hermval, hermval2d, hermval3d, hermgrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def hermgrid2d(x, y, c):
"""
Evaluate a 2-D Hermite series on the Cartesian product of x and y.
This function returns the values:
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_j(b)
where the points `(a, b)` consist of all pairs formed by taking
`a` from `x` and `b` from `y`. The resulting points form a grid with
`x` in the first dimension and `y` in the second.
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars. In either
case, either `x` and `y` or their elements must support multiplication
and addition both with themselves and with the elements of `c`.
If `c` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points in the
Cartesian product of `x` and `y`. If `x` or `y` is a list or
tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree i,j are contained in ``c[i,j]``. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points in the Cartesian
product of `x` and `y`.
See Also
--------
hermval, hermval2d, hermval3d, hermgrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(hermval, c, x, y) | Evaluate a 2-D Hermite series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_j(b) where the points `(a, b)` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- hermval, hermval2d, hermval3d, hermgrid3d Notes ----- .. versionadded:: 1.7.0 |
169,751 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermval(x, c, tensor=True):
"""
Evaluate an Hermite series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
hermval2d, hermgrid2d, hermval3d, hermgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.hermite import hermval
>>> coef = [1,2,3]
>>> hermval(1, coef)
11.0
>>> hermval([[1,2],[3,4]], coef)
array([[ 11., 51.],
[115., 203.]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
x2 = x*2
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - c1*(2*(nd - 1))
c1 = tmp + c1*x2
return c0 + c1*x2
The provided code snippet includes necessary dependencies for implementing the `hermval3d` function. Write a Python function `def hermval3d(x, y, z, c)` to solve the following problem:
Evaluate a 3-D Hermite series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * H_i(x) * H_j(y) * H_k(z) The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- hermval, hermval2d, hermgrid2d, hermgrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def hermval3d(x, y, z, c):
"""
Evaluate a 3-D Hermite series at points (x, y, z).
This function returns the values:
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * H_i(x) * H_j(y) * H_k(z)
The parameters `x`, `y`, and `z` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
`x`, `y`, and `z` or their elements must support multiplication and
addition both with themselves and with the elements of `c`.
If `c` has fewer than 3 dimensions, ones are implicitly appended to its
shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape.
Parameters
----------
x, y, z : array_like, compatible object
The three dimensional series is evaluated at the points
`(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If
any of `x`, `y`, or `z` is a list or tuple, it is first converted
to an ndarray, otherwise it is left unchanged and if it isn't an
ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
greater than 3 the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the multidimensional polynomial on points formed with
triples of corresponding values from `x`, `y`, and `z`.
See Also
--------
hermval, hermval2d, hermgrid2d, hermgrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(hermval, c, x, y, z) | Evaluate a 3-D Hermite series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * H_i(x) * H_j(y) * H_k(z) The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- hermval, hermval2d, hermgrid2d, hermgrid3d Notes ----- .. versionadded:: 1.7.0 |
169,752 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermval(x, c, tensor=True):
"""
Evaluate an Hermite series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
hermval2d, hermgrid2d, hermval3d, hermgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.hermite import hermval
>>> coef = [1,2,3]
>>> hermval(1, coef)
11.0
>>> hermval([[1,2],[3,4]], coef)
array([[ 11., 51.],
[115., 203.]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
x2 = x*2
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - c1*(2*(nd - 1))
c1 = tmp + c1*x2
return c0 + c1*x2
The provided code snippet includes necessary dependencies for implementing the `hermgrid3d` function. Write a Python function `def hermgrid3d(x, y, z, c)` to solve the following problem:
Evaluate a 3-D Hermite series on the Cartesian product of x, y, and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * H_i(a) * H_j(b) * H_k(c) where the points `(a, b, c)` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- hermval, hermval2d, hermgrid2d, hermval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def hermgrid3d(x, y, z, c):
"""
Evaluate a 3-D Hermite series on the Cartesian product of x, y, and z.
This function returns the values:
.. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * H_i(a) * H_j(b) * H_k(c)
where the points `(a, b, c)` consist of all triples formed by taking
`a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
a grid with `x` in the first dimension, `y` in the second, and `z` in
the third.
The parameters `x`, `y`, and `z` are converted to arrays only if they
are tuples or a lists, otherwise they are treated as a scalars. In
either case, either `x`, `y`, and `z` or their elements must support
multiplication and addition both with themselves and with the elements
of `c`.
If `c` has fewer than three dimensions, ones are implicitly appended to
its shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape + y.shape + z.shape.
Parameters
----------
x, y, z : array_like, compatible objects
The three dimensional series is evaluated at the points in the
Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a
list or tuple, it is first converted to an ndarray, otherwise it is
left unchanged and, if it isn't an ndarray, it is treated as a
scalar.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree i,j are contained in ``c[i,j]``. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points in the Cartesian
product of `x` and `y`.
See Also
--------
hermval, hermval2d, hermgrid2d, hermval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(hermval, c, x, y, z) | Evaluate a 3-D Hermite series on the Cartesian product of x, y, and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * H_i(a) * H_j(b) * H_k(c) where the points `(a, b, c)` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- hermval, hermval2d, hermgrid2d, hermval3d Notes ----- .. versionadded:: 1.7.0 |
169,753 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = H_i(x),
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Hermite polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
array ``V = hermvander(x, n)``, then ``np.dot(V, c)`` and
``hermval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Hermite series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo-Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Hermite polynomial. The dtype will be the same as
the converted `x`.
Examples
--------
>>> from numpy.polynomial.hermite import hermvander
>>> x = np.array([-1, 0, 1])
>>> hermvander(x, 3)
array([[ 1., -2., 2., 4.],
[ 1., 0., -2., -0.],
[ 1., 2., 2., -4.]])
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
if ideg > 0:
x2 = x*2
v[1] = x2
for i in range(2, ideg + 1):
v[i] = (v[i-1]*x2 - v[i-2]*(2*(i - 1)))
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `hermvander2d` function. Write a Python function `def hermvander2d(x, y, deg)` to solve the following problem:
Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y)`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = H_i(x) * H_j(y), where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of `V` index the points `(x, y)` and the last index encodes the degrees of the Hermite polynomials. If ``V = hermvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``hermval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Hermite series of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- hermvander, hermvander3d, hermval2d, hermval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def hermvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y)`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (deg[1] + 1)*i + j] = H_i(x) * H_j(y),
where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
`V` index the points `(x, y)` and the last index encodes the degrees of
the Hermite polynomials.
If ``V = hermvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
correspond to the elements of a 2-D coefficient array `c` of shape
(xdeg + 1, ydeg + 1) in the order
.. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
and ``np.dot(V, c.flat)`` and ``hermval2d(x, y, c)`` will be the same
up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 2-D Hermite
series of the same degrees and sample points.
Parameters
----------
x, y : array_like
Arrays of point coordinates, all of the same shape. The dtypes
will be converted to either float64 or complex128 depending on
whether any of the elements are complex. Scalars are converted to 1-D
arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg].
Returns
-------
vander2d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same
as the converted `x` and `y`.
See Also
--------
hermvander, hermvander3d, hermval2d, hermval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((hermvander, hermvander), (x, y), deg) | Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y)`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = H_i(x) * H_j(y), where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of `V` index the points `(x, y)` and the last index encodes the degrees of the Hermite polynomials. If ``V = hermvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``hermval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Hermite series of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- hermvander, hermvander3d, hermval2d, hermval3d Notes ----- .. versionadded:: 1.7.0 |
169,754 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = H_i(x),
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Hermite polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
array ``V = hermvander(x, n)``, then ``np.dot(V, c)`` and
``hermval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Hermite series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo-Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Hermite polynomial. The dtype will be the same as
the converted `x`.
Examples
--------
>>> from numpy.polynomial.hermite import hermvander
>>> x = np.array([-1, 0, 1])
>>> hermvander(x, 3)
array([[ 1., -2., 2., 4.],
[ 1., 0., -2., -0.],
[ 1., 2., 2., -4.]])
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
if ideg > 0:
x2 = x*2
v[1] = x2
for i in range(2, ideg + 1):
v[i] = (v[i-1]*x2 - v[i-2]*(2*(i - 1)))
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `hermvander3d` function. Write a Python function `def hermvander3d(x, y, z, deg)` to solve the following problem:
Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = H_i(x)*H_j(y)*H_k(z), where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading indices of `V` index the points `(x, y, z)` and the last index encodes the degrees of the Hermite polynomials. If ``V = hermvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``hermval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Hermite series of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- hermvander, hermvander3d, hermval2d, hermval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def hermvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
then The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = H_i(x)*H_j(y)*H_k(z),
where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading
indices of `V` index the points `(x, y, z)` and the last index encodes
the degrees of the Hermite polynomials.
If ``V = hermvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
of `V` correspond to the elements of a 3-D coefficient array `c` of
shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
.. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
and ``np.dot(V, c.flat)`` and ``hermval3d(x, y, z, c)`` will be the
same up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 3-D Hermite
series of the same degrees and sample points.
Parameters
----------
x, y, z : array_like
Arrays of point coordinates, all of the same shape. The dtypes will
be converted to either float64 or complex128 depending on whether
any of the elements are complex. Scalars are converted to 1-D
arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg, z_deg].
Returns
-------
vander3d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will
be the same as the converted `x`, `y`, and `z`.
See Also
--------
hermvander, hermvander3d, hermval2d, hermval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((hermvander, hermvander, hermvander), (x, y, z), deg) | Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = H_i(x)*H_j(y)*H_k(z), where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading indices of `V` index the points `(x, y, z)` and the last index encodes the degrees of the Hermite polynomials. If ``V = hermvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``hermval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Hermite series of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- hermvander, hermvander3d, hermval2d, hermval3d Notes ----- .. versionadded:: 1.7.0 |
169,755 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = H_i(x),
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Hermite polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
array ``V = hermvander(x, n)``, then ``np.dot(V, c)`` and
``hermval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Hermite series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo-Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Hermite polynomial. The dtype will be the same as
the converted `x`.
Examples
--------
>>> from numpy.polynomial.hermite import hermvander
>>> x = np.array([-1, 0, 1])
>>> hermvander(x, 3)
array([[ 1., -2., 2., 4.],
[ 1., 0., -2., -0.],
[ 1., 2., 2., -4.]])
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
if ideg > 0:
x2 = x*2
v[1] = x2
for i in range(2, ideg + 1):
v[i] = (v[i-1]*x2 - v[i-2]*(2*(i - 1)))
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `hermfit` function. Write a Python function `def hermfit(x, y, deg, rcond=None, full=False, w=None)` to solve the following problem:
Least squares fit of Hermite series to data. Return the coefficients of a Hermite series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x), where `n` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the weight ``w[i]`` applies to the unsquared residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. When using inverse-variance weighting, use ``w[i] = 1/sigma(y[i])``. The default value is None. Returns ------- coef : ndarray, shape (M,) or (M, K) Hermite coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`. [residuals, rank, singular_values, rcond] : list These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the numerical rank of the scaled Vandermonde matrix - singular_values -- singular values of the scaled Vandermonde matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if ``full == False``. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- numpy.polynomial.chebyshev.chebfit numpy.polynomial.legendre.legfit numpy.polynomial.laguerre.lagfit numpy.polynomial.polynomial.polyfit numpy.polynomial.hermite_e.hermefit hermval : Evaluates a Hermite series. hermvander : Vandermonde matrix of Hermite series. hermweight : Hermite weight function numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Hermite series `p` that minimizes the sum of the weighted squared errors .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, where the :math:`w_j` are the weights. This problem is solved by setting up the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the coefficients to be solved for, `w` are the weights, `y` are the observed values. This equation is then solved using the singular value decomposition of `V`. If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Hermite series are probably most useful when the data can be approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the Hermite weight. In that case the weight ``sqrt(w(x[i]))`` should be used together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is available as `hermweight`. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples -------- >>> from numpy.polynomial.hermite import hermfit, hermval >>> x = np.linspace(-10, 10) >>> err = np.random.randn(len(x))/10 >>> y = hermval(x, [1, 2, 3]) + err >>> hermfit(x, y, 2) array([1.0218, 1.9986, 2.9999]) # may vary
Here is the function:
def hermfit(x, y, deg, rcond=None, full=False, w=None):
"""
Least squares fit of Hermite series to data.
Return the coefficients of a Hermite series of degree `deg` that is the
least squares fit to the data values `y` given at points `x`. If `y` is
1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
fits are done, one for each column of `y`, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form
.. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x),
where `n` is `deg`.
Parameters
----------
x : array_like, shape (M,)
x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,) or (M, K)
y-coordinates of the sample points. Several data sets of sample
points sharing the same x-coordinates can be fitted at once by
passing in a 2D-array that contains one dataset per column.
deg : int or 1-D array_like
Degree(s) of the fitting polynomials. If `deg` is a single integer
all terms up to and including the `deg`'th term are included in the
fit. For NumPy versions >= 1.11.0 a list of integers specifying the
degrees of the terms to include may be used instead.
rcond : float, optional
Relative condition number of the fit. Singular values smaller than
this relative to the largest singular value will be ignored. The
default value is len(x)*eps, where eps is the relative precision of
the float type, about 2e-16 in most cases.
full : bool, optional
Switch determining nature of return value. When it is False (the
default) just the coefficients are returned, when True diagnostic
information from the singular value decomposition is also returned.
w : array_like, shape (`M`,), optional
Weights. If not None, the weight ``w[i]`` applies to the unsquared
residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
chosen so that the errors of the products ``w[i]*y[i]`` all have the
same variance. When using inverse-variance weighting, use
``w[i] = 1/sigma(y[i])``. The default value is None.
Returns
-------
coef : ndarray, shape (M,) or (M, K)
Hermite coefficients ordered from low to high. If `y` was 2-D,
the coefficients for the data in column k of `y` are in column
`k`.
[residuals, rank, singular_values, rcond] : list
These values are only returned if ``full == True``
- residuals -- sum of squared residuals of the least squares fit
- rank -- the numerical rank of the scaled Vandermonde matrix
- singular_values -- singular values of the scaled Vandermonde matrix
- rcond -- value of `rcond`.
For more details, see `numpy.linalg.lstsq`.
Warns
-----
RankWarning
The rank of the coefficient matrix in the least-squares fit is
deficient. The warning is only raised if ``full == False``. The
warnings can be turned off by
>>> import warnings
>>> warnings.simplefilter('ignore', np.RankWarning)
See Also
--------
numpy.polynomial.chebyshev.chebfit
numpy.polynomial.legendre.legfit
numpy.polynomial.laguerre.lagfit
numpy.polynomial.polynomial.polyfit
numpy.polynomial.hermite_e.hermefit
hermval : Evaluates a Hermite series.
hermvander : Vandermonde matrix of Hermite series.
hermweight : Hermite weight function
numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes
-----
The solution is the coefficients of the Hermite series `p` that
minimizes the sum of the weighted squared errors
.. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
where the :math:`w_j` are the weights. This problem is solved by
setting up the (typically) overdetermined matrix equation
.. math:: V(x) * c = w * y,
where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
coefficients to be solved for, `w` are the weights, `y` are the
observed values. This equation is then solved using the singular value
decomposition of `V`.
If some of the singular values of `V` are so small that they are
neglected, then a `RankWarning` will be issued. This means that the
coefficient values may be poorly determined. Using a lower order fit
will usually get rid of the warning. The `rcond` parameter can also be
set to a value smaller than its default, but the resulting fit may be
spurious and have large contributions from roundoff error.
Fits using Hermite series are probably most useful when the data can be
approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the Hermite
weight. In that case the weight ``sqrt(w(x[i]))`` should be used
together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is
available as `hermweight`.
References
----------
.. [1] Wikipedia, "Curve fitting",
https://en.wikipedia.org/wiki/Curve_fitting
Examples
--------
>>> from numpy.polynomial.hermite import hermfit, hermval
>>> x = np.linspace(-10, 10)
>>> err = np.random.randn(len(x))/10
>>> y = hermval(x, [1, 2, 3]) + err
>>> hermfit(x, y, 2)
array([1.0218, 1.9986, 2.9999]) # may vary
"""
return pu._fit(hermvander, x, y, deg, rcond, full, w) | Least squares fit of Hermite series to data. Return the coefficients of a Hermite series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x), where `n` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the weight ``w[i]`` applies to the unsquared residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. When using inverse-variance weighting, use ``w[i] = 1/sigma(y[i])``. The default value is None. Returns ------- coef : ndarray, shape (M,) or (M, K) Hermite coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`. [residuals, rank, singular_values, rcond] : list These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the numerical rank of the scaled Vandermonde matrix - singular_values -- singular values of the scaled Vandermonde matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if ``full == False``. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- numpy.polynomial.chebyshev.chebfit numpy.polynomial.legendre.legfit numpy.polynomial.laguerre.lagfit numpy.polynomial.polynomial.polyfit numpy.polynomial.hermite_e.hermefit hermval : Evaluates a Hermite series. hermvander : Vandermonde matrix of Hermite series. hermweight : Hermite weight function numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Hermite series `p` that minimizes the sum of the weighted squared errors .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, where the :math:`w_j` are the weights. This problem is solved by setting up the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the coefficients to be solved for, `w` are the weights, `y` are the observed values. This equation is then solved using the singular value decomposition of `V`. If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Hermite series are probably most useful when the data can be approximated by ``sqrt(w(x)) * p(x)``, where `w(x)` is the Hermite weight. In that case the weight ``sqrt(w(x[i]))`` should be used together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is available as `hermweight`. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples -------- >>> from numpy.polynomial.hermite import hermfit, hermval >>> x = np.linspace(-10, 10) >>> err = np.random.randn(len(x))/10 >>> y = hermval(x, [1, 2, 3]) + err >>> hermfit(x, y, 2) array([1.0218, 1.9986, 2.9999]) # may vary |
169,756 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Hermite basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded:: 1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-.5*c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
scl = np.multiply.accumulate(scl)[::-1]
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.sqrt(.5*np.arange(1, n))
bot[...] = top
mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
return mat
The provided code snippet includes necessary dependencies for implementing the `hermroots` function. Write a Python function `def hermroots(c)` to solve the following problem:
Compute the roots of a Hermite series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * H_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex. See Also -------- numpy.polynomial.polynomial.polyroots numpy.polynomial.legendre.legroots numpy.polynomial.laguerre.lagroots numpy.polynomial.chebyshev.chebroots numpy.polynomial.hermite_e.hermeroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method. The Hermite series basis polynomials aren't powers of `x` so the results of this function may seem unintuitive. Examples -------- >>> from numpy.polynomial.hermite import hermroots, hermfromroots >>> coef = hermfromroots([-1, 0, 1]) >>> coef array([0. , 0.25 , 0. , 0.125]) >>> hermroots(coef) array([-1.00000000e+00, -1.38777878e-17, 1.00000000e+00])
Here is the function:
def hermroots(c):
"""
Compute the roots of a Hermite series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * H_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
numpy.polynomial.polynomial.polyroots
numpy.polynomial.legendre.legroots
numpy.polynomial.laguerre.lagroots
numpy.polynomial.chebyshev.chebroots
numpy.polynomial.hermite_e.hermeroots
Notes
-----
The root estimates are obtained as the eigenvalues of the companion
matrix, Roots far from the origin of the complex plane may have large
errors due to the numerical instability of the series for such
values. Roots with multiplicity greater than 1 will also show larger
errors as the value of the series near such points is relatively
insensitive to errors in the roots. Isolated roots near the origin can
be improved by a few iterations of Newton's method.
The Hermite series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.hermite import hermroots, hermfromroots
>>> coef = hermfromroots([-1, 0, 1])
>>> coef
array([0. , 0.25 , 0. , 0.125])
>>> hermroots(coef)
array([-1.00000000e+00, -1.38777878e-17, 1.00000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-.5*c[0]/c[1]])
# rotated companion matrix reduces error
m = hermcompanion(c)[::-1,::-1]
r = la.eigvals(m)
r.sort()
return r | Compute the roots of a Hermite series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * H_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex. See Also -------- numpy.polynomial.polynomial.polyroots numpy.polynomial.legendre.legroots numpy.polynomial.laguerre.lagroots numpy.polynomial.chebyshev.chebroots numpy.polynomial.hermite_e.hermeroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method. The Hermite series basis polynomials aren't powers of `x` so the results of this function may seem unintuitive. Examples -------- >>> from numpy.polynomial.hermite import hermroots, hermfromroots >>> coef = hermfromroots([-1, 0, 1]) >>> coef array([0. , 0.25 , 0. , 0.125]) >>> hermroots(coef) array([-1.00000000e+00, -1.38777878e-17, 1.00000000e+00]) |
169,757 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def hermcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is an Hermite basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Hermite series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded:: 1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-.5*c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1))))
scl = np.multiply.accumulate(scl)[::-1]
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = np.sqrt(.5*np.arange(1, n))
bot[...] = top
mat[:, -1] -= scl*c[:-1]/(2.0*c[-1])
return mat
def _normed_hermite_n(x, n):
"""
Evaluate a normalized Hermite polynomial.
Compute the value of the normalized Hermite polynomial of degree ``n``
at the points ``x``.
Parameters
----------
x : ndarray of double.
Points at which to evaluate the function
n : int
Degree of the normalized Hermite function to be evaluated.
Returns
-------
values : ndarray
The shape of the return value is described above.
Notes
-----
.. versionadded:: 1.10.0
This function is needed for finding the Gauss points and integration
weights for high degrees. The values of the standard Hermite functions
overflow when n >= 207.
"""
if n == 0:
return np.full(x.shape, 1/np.sqrt(np.sqrt(np.pi)))
c0 = 0.
c1 = 1./np.sqrt(np.sqrt(np.pi))
nd = float(n)
for i in range(n - 1):
tmp = c0
c0 = -c1*np.sqrt((nd - 1.)/nd)
c1 = tmp + c1*x*np.sqrt(2./nd)
nd = nd - 1.0
return c0 + c1*x*np.sqrt(2)
The provided code snippet includes necessary dependencies for implementing the `hermgauss` function. Write a Python function `def hermgauss(deg)` to solve the following problem:
Gauss-Hermite quadrature. Computes the sample points and weights for Gauss-Hermite quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]` with the weight function :math:`f(x) = \\exp(-x^2)`. Parameters ---------- deg : int Number of sample points and weights. It must be >= 1. Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights. Notes ----- .. versionadded:: 1.7.0 The results have only been tested up to degree 100, higher degrees may be problematic. The weights are determined by using the fact that .. math:: w_k = c / (H'_n(x_k) * H_{n-1}(x_k)) where :math:`c` is a constant independent of :math:`k` and :math:`x_k` is the k'th root of :math:`H_n`, and then scaling the results to get the right value when integrating 1.
Here is the function:
def hermgauss(deg):
"""
Gauss-Hermite quadrature.
Computes the sample points and weights for Gauss-Hermite quadrature.
These sample points and weights will correctly integrate polynomials of
degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]`
with the weight function :math:`f(x) = \\exp(-x^2)`.
Parameters
----------
deg : int
Number of sample points and weights. It must be >= 1.
Returns
-------
x : ndarray
1-D ndarray containing the sample points.
y : ndarray
1-D ndarray containing the weights.
Notes
-----
.. versionadded:: 1.7.0
The results have only been tested up to degree 100, higher degrees may
be problematic. The weights are determined by using the fact that
.. math:: w_k = c / (H'_n(x_k) * H_{n-1}(x_k))
where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
is the k'th root of :math:`H_n`, and then scaling the results to get
the right value when integrating 1.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg <= 0:
raise ValueError("deg must be a positive integer")
# first approximation of roots. We use the fact that the companion
# matrix is symmetric in this case in order to obtain better zeros.
c = np.array([0]*deg + [1], dtype=np.float64)
m = hermcompanion(c)
x = la.eigvalsh(m)
# improve roots by one application of Newton
dy = _normed_hermite_n(x, ideg)
df = _normed_hermite_n(x, ideg - 1) * np.sqrt(2*ideg)
x -= dy/df
# compute the weights. We scale the factor to avoid possible numerical
# overflow.
fm = _normed_hermite_n(x, ideg - 1)
fm /= np.abs(fm).max()
w = 1/(fm * fm)
# for Hermite we can also symmetrize
w = (w + w[::-1])/2
x = (x - x[::-1])/2
# scale w to get the right value
w *= np.sqrt(np.pi) / w.sum()
return x, w | Gauss-Hermite quadrature. Computes the sample points and weights for Gauss-Hermite quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]` with the weight function :math:`f(x) = \\exp(-x^2)`. Parameters ---------- deg : int Number of sample points and weights. It must be >= 1. Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights. Notes ----- .. versionadded:: 1.7.0 The results have only been tested up to degree 100, higher degrees may be problematic. The weights are determined by using the fact that .. math:: w_k = c / (H'_n(x_k) * H_{n-1}(x_k)) where :math:`c` is a constant independent of :math:`k` and :math:`x_k` is the k'th root of :math:`H_n`, and then scaling the results to get the right value when integrating 1. |
169,758 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `hermweight` function. Write a Python function `def hermweight(x)` to solve the following problem:
Weight function of the Hermite polynomials. The weight function is :math:`\\exp(-x^2)` and the interval of integration is :math:`[-\\inf, \\inf]`. the Hermite polynomials are orthogonal, but not normalized, with respect to this weight function. Parameters ---------- x : array_like Values at which the weight function will be computed. Returns ------- w : ndarray The weight function at `x`. Notes ----- .. versionadded:: 1.7.0
Here is the function:
def hermweight(x):
"""
Weight function of the Hermite polynomials.
The weight function is :math:`\\exp(-x^2)` and the interval of
integration is :math:`[-\\inf, \\inf]`. the Hermite polynomials are
orthogonal, but not normalized, with respect to this weight function.
Parameters
----------
x : array_like
Values at which the weight function will be computed.
Returns
-------
w : ndarray
The weight function at `x`.
Notes
-----
.. versionadded:: 1.7.0
"""
w = np.exp(-x**2)
return w | Weight function of the Hermite polynomials. The weight function is :math:`\\exp(-x^2)` and the interval of integration is :math:`[-\\inf, \\inf]`. the Hermite polynomials are orthogonal, but not normalized, with respect to this weight function. Parameters ---------- x : array_like Values at which the weight function will be computed. Returns ------- w : ndarray The weight function at `x`. Notes ----- .. versionadded:: 1.7.0 |
169,759 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagadd(c1, c2):
"""
Add one Laguerre series to another.
Returns the sum of two Laguerre series `c1` + `c2`. The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Laguerre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the Laguerre series of their sum.
See Also
--------
lagsub, lagmulx, lagmul, lagdiv, lagpow
Notes
-----
Unlike multiplication, division, etc., the sum of two Laguerre series
is a Laguerre series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."
Examples
--------
>>> from numpy.polynomial.laguerre import lagadd
>>> lagadd([1, 2, 3], [1, 2, 3, 4])
array([2., 4., 6., 4.])
"""
return pu._add(c1, c2)
def lagmulx(c):
"""Multiply a Laguerre series by x.
Multiply the Laguerre series `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of Laguerre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
See Also
--------
lagadd, lagsub, lagmul, lagdiv, lagpow
Notes
-----
The multiplication uses the recursion relationship for Laguerre
polynomials in the form
.. math::
xP_i(x) = (-(i + 1)*P_{i + 1}(x) + (2i + 1)P_{i}(x) - iP_{i - 1}(x))
Examples
--------
>>> from numpy.polynomial.laguerre import lagmulx
>>> lagmulx([1, 2, 3])
array([-1., -1., 11., -9.])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]
prd[1] = -c[0]
for i in range(1, len(c)):
prd[i + 1] = -c[i]*(i + 1)
prd[i] += c[i]*(2*i + 1)
prd[i - 1] -= c[i]*i
return prd
The provided code snippet includes necessary dependencies for implementing the `poly2lag` function. Write a Python function `def poly2lag(pol)` to solve the following problem:
poly2lag(pol) Convert a polynomial to a Laguerre series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Laguerre series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Laguerre series. See Also -------- lag2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.laguerre import poly2lag >>> poly2lag(np.arange(4)) array([ 23., -63., 58., -18.])
Here is the function:
def poly2lag(pol):
"""
poly2lag(pol)
Convert a polynomial to a Laguerre series.
Convert an array representing the coefficients of a polynomial (relative
to the "standard" basis) ordered from lowest degree to highest, to an
array of the coefficients of the equivalent Laguerre series, ordered
from lowest to highest degree.
Parameters
----------
pol : array_like
1-D array containing the polynomial coefficients
Returns
-------
c : ndarray
1-D array containing the coefficients of the equivalent Laguerre
series.
See Also
--------
lag2poly
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy.polynomial.laguerre import poly2lag
>>> poly2lag(np.arange(4))
array([ 23., -63., 58., -18.])
"""
[pol] = pu.as_series([pol])
res = 0
for p in pol[::-1]:
res = lagadd(lagmulx(res), p)
return res | poly2lag(pol) Convert a polynomial to a Laguerre series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Laguerre series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Laguerre series. See Also -------- lag2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.laguerre import poly2lag >>> poly2lag(np.arange(4)) array([ 23., -63., 58., -18.]) |
169,760 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyadd(c1, c2):
"""
Add one polynomial to another.
Returns the sum of two polynomials `c1` + `c2`. The arguments are
sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to high.
Returns
-------
out : ndarray
The coefficient array representing their sum.
See Also
--------
polysub, polymulx, polymul, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> sum = P.polyadd(c1,c2); sum
array([4., 4., 4.])
>>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2)
28.0
"""
return pu._add(c1, c2)
def polysub(c1, c2):
"""
Subtract one polynomial from another.
Returns the difference of two polynomials `c1` - `c2`. The arguments
are sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of coefficients representing their difference.
See Also
--------
polyadd, polymulx, polymul, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polysub(c1,c2)
array([-2., 0., 2.])
>>> P.polysub(c2,c1) # -P.polysub(c1,c2)
array([ 2., 0., -2.])
"""
return pu._sub(c1, c2)
def polymulx(c):
"""Multiply a polynomial by x.
Multiply the polynomial `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
See Also
--------
polyadd, polysub, polymul, polydiv, polypow
Notes
-----
.. versionadded:: 1.5.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]*0
prd[1:] = c
return prd
The provided code snippet includes necessary dependencies for implementing the `lag2poly` function. Write a Python function `def lag2poly(c)` to solve the following problem:
Convert a Laguerre series to a polynomial. Convert an array representing the coefficients of a Laguerre series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- c : array_like 1-D array containing the Laguerre series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2lag Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.laguerre import lag2poly >>> lag2poly([ 23., -63., 58., -18.]) array([0., 1., 2., 3.])
Here is the function:
def lag2poly(c):
"""
Convert a Laguerre series to a polynomial.
Convert an array representing the coefficients of a Laguerre series,
ordered from lowest degree to highest, to an array of the coefficients
of the equivalent polynomial (relative to the "standard" basis) ordered
from lowest to highest degree.
Parameters
----------
c : array_like
1-D array containing the Laguerre series coefficients, ordered
from lowest order term to highest.
Returns
-------
pol : ndarray
1-D array containing the coefficients of the equivalent polynomial
(relative to the "standard" basis) ordered from lowest order term
to highest.
See Also
--------
poly2lag
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy.polynomial.laguerre import lag2poly
>>> lag2poly([ 23., -63., 58., -18.])
array([0., 1., 2., 3.])
"""
from .polynomial import polyadd, polysub, polymulx
[c] = pu.as_series([c])
n = len(c)
if n == 1:
return c
else:
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], (c1*(i - 1))/i)
c1 = polyadd(tmp, polysub((2*i - 1)*c1, polymulx(c1))/i)
return polyadd(c0, polysub(c1, polymulx(c1))) | Convert a Laguerre series to a polynomial. Convert an array representing the coefficients of a Laguerre series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- c : array_like 1-D array containing the Laguerre series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2lag Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy.polynomial.laguerre import lag2poly >>> lag2poly([ 23., -63., 58., -18.]) array([0., 1., 2., 3.]) |
169,761 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagline(off, scl):
"""
Laguerre series whose graph is a straight line.
Parameters
----------
off, scl : scalars
The specified line is given by ``off + scl*x``.
Returns
-------
y : ndarray
This module's representation of the Laguerre series for
``off + scl*x``.
See Also
--------
numpy.polynomial.polynomial.polyline
numpy.polynomial.chebyshev.chebline
numpy.polynomial.legendre.legline
numpy.polynomial.hermite.hermline
numpy.polynomial.hermite_e.hermeline
Examples
--------
>>> from numpy.polynomial.laguerre import lagline, lagval
>>> lagval(0,lagline(3, 2))
3.0
>>> lagval(1,lagline(3, 2))
5.0
"""
if scl != 0:
return np.array([off + scl, -scl])
else:
return np.array([off])
def lagmul(c1, c2):
"""
Multiply one Laguerre series by another.
Returns the product of two Laguerre series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Laguerre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Laguerre series coefficients representing their product.
See Also
--------
lagadd, lagsub, lagmulx, lagdiv, lagpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Laguerre polynomial basis set. Thus, to express
the product as a Laguerre series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial.laguerre import lagmul
>>> lagmul([1, 2, 3], [0, 1, 2])
array([ 8., -13., 38., -51., 36.])
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = lagsub(c[-i]*xs, (c1*(nd - 1))/nd)
c1 = lagadd(tmp, lagsub((2*nd - 1)*c1, lagmulx(c1))/nd)
return lagadd(c0, lagsub(c1, lagmulx(c1)))
The provided code snippet includes necessary dependencies for implementing the `lagfromroots` function. Write a Python function `def lagfromroots(roots)` to solve the following problem:
Generate a Laguerre series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in Laguerre form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x) The coefficient of the last term is not generally 1 for monic polynomials in Laguerre form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below). See Also -------- numpy.polynomial.polynomial.polyfromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.hermite.hermfromroots numpy.polynomial.hermite_e.hermefromroots Examples -------- >>> from numpy.polynomial.laguerre import lagfromroots, lagval >>> coef = lagfromroots((-1, 0, 1)) >>> lagval((-1, 0, 1), coef) array([0., 0., 0.]) >>> coef = lagfromroots((-1j, 1j)) >>> lagval((-1j, 1j), coef) array([0.+0.j, 0.+0.j])
Here is the function:
def lagfromroots(roots):
"""
Generate a Laguerre series with given roots.
The function returns the coefficients of the polynomial
.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
in Laguerre form, where the `r_n` are the roots specified in `roots`.
If a zero has multiplicity n, then it must appear in `roots` n times.
For instance, if 2 is a root of multiplicity three and 3 is a root of
multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
roots can appear in any order.
If the returned coefficients are `c`, then
.. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x)
The coefficient of the last term is not generally 1 for monic
polynomials in Laguerre form.
Parameters
----------
roots : array_like
Sequence containing the roots.
Returns
-------
out : ndarray
1-D array of coefficients. If all roots are real then `out` is a
real array, if some of the roots are complex, then `out` is complex
even if all the coefficients in the result are real (see Examples
below).
See Also
--------
numpy.polynomial.polynomial.polyfromroots
numpy.polynomial.legendre.legfromroots
numpy.polynomial.chebyshev.chebfromroots
numpy.polynomial.hermite.hermfromroots
numpy.polynomial.hermite_e.hermefromroots
Examples
--------
>>> from numpy.polynomial.laguerre import lagfromroots, lagval
>>> coef = lagfromroots((-1, 0, 1))
>>> lagval((-1, 0, 1), coef)
array([0., 0., 0.])
>>> coef = lagfromroots((-1j, 1j))
>>> lagval((-1j, 1j), coef)
array([0.+0.j, 0.+0.j])
"""
return pu._fromroots(lagline, lagmul, roots) | Generate a Laguerre series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in Laguerre form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x) The coefficient of the last term is not generally 1 for monic polynomials in Laguerre form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below). See Also -------- numpy.polynomial.polynomial.polyfromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.hermite.hermfromroots numpy.polynomial.hermite_e.hermefromroots Examples -------- >>> from numpy.polynomial.laguerre import lagfromroots, lagval >>> coef = lagfromroots((-1, 0, 1)) >>> lagval((-1, 0, 1), coef) array([0., 0., 0.]) >>> coef = lagfromroots((-1j, 1j)) >>> lagval((-1j, 1j), coef) array([0.+0.j, 0.+0.j]) |
169,762 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagmul(c1, c2):
"""
Multiply one Laguerre series by another.
Returns the product of two Laguerre series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Laguerre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Laguerre series coefficients representing their product.
See Also
--------
lagadd, lagsub, lagmulx, lagdiv, lagpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Laguerre polynomial basis set. Thus, to express
the product as a Laguerre series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial.laguerre import lagmul
>>> lagmul([1, 2, 3], [0, 1, 2])
array([ 8., -13., 38., -51., 36.])
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = lagsub(c[-i]*xs, (c1*(nd - 1))/nd)
c1 = lagadd(tmp, lagsub((2*nd - 1)*c1, lagmulx(c1))/nd)
return lagadd(c0, lagsub(c1, lagmulx(c1)))
The provided code snippet includes necessary dependencies for implementing the `lagdiv` function. Write a Python function `def lagdiv(c1, c2)` to solve the following problem:
Divide one Laguerre series by another. Returns the quotient-with-remainder of two Laguerre series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Laguerre series coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of Laguerre series coefficients representing the quotient and remainder. See Also -------- lagadd, lagsub, lagmulx, lagmul, lagpow Notes ----- In general, the (polynomial) division of one Laguerre series by another results in quotient and remainder terms that are not in the Laguerre polynomial basis set. Thus, to express these results as a Laguerre series, it is necessary to "reproject" the results onto the Laguerre basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial.laguerre import lagdiv >>> lagdiv([ 8., -13., 38., -51., 36.], [0, 1, 2]) (array([1., 2., 3.]), array([0.])) >>> lagdiv([ 9., -12., 38., -51., 36.], [0, 1, 2]) (array([1., 2., 3.]), array([1., 1.]))
Here is the function:
def lagdiv(c1, c2):
"""
Divide one Laguerre series by another.
Returns the quotient-with-remainder of two Laguerre series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Laguerre series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarrays
Of Laguerre series coefficients representing the quotient and
remainder.
See Also
--------
lagadd, lagsub, lagmulx, lagmul, lagpow
Notes
-----
In general, the (polynomial) division of one Laguerre series by another
results in quotient and remainder terms that are not in the Laguerre
polynomial basis set. Thus, to express these results as a Laguerre
series, it is necessary to "reproject" the results onto the Laguerre
basis set, which may produce "unintuitive" (but correct) results; see
Examples section below.
Examples
--------
>>> from numpy.polynomial.laguerre import lagdiv
>>> lagdiv([ 8., -13., 38., -51., 36.], [0, 1, 2])
(array([1., 2., 3.]), array([0.]))
>>> lagdiv([ 9., -12., 38., -51., 36.], [0, 1, 2])
(array([1., 2., 3.]), array([1., 1.]))
"""
return pu._div(lagmul, c1, c2) | Divide one Laguerre series by another. Returns the quotient-with-remainder of two Laguerre series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Laguerre series coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of Laguerre series coefficients representing the quotient and remainder. See Also -------- lagadd, lagsub, lagmulx, lagmul, lagpow Notes ----- In general, the (polynomial) division of one Laguerre series by another results in quotient and remainder terms that are not in the Laguerre polynomial basis set. Thus, to express these results as a Laguerre series, it is necessary to "reproject" the results onto the Laguerre basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial.laguerre import lagdiv >>> lagdiv([ 8., -13., 38., -51., 36.], [0, 1, 2]) (array([1., 2., 3.]), array([0.])) >>> lagdiv([ 9., -12., 38., -51., 36.], [0, 1, 2]) (array([1., 2., 3.]), array([1., 1.])) |
169,763 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagmul(c1, c2):
"""
Multiply one Laguerre series by another.
Returns the product of two Laguerre series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Laguerre series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Laguerre series coefficients representing their product.
See Also
--------
lagadd, lagsub, lagmulx, lagdiv, lagpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Laguerre polynomial basis set. Thus, to express
the product as a Laguerre series, it is necessary to "reproject" the
product onto said basis set, which may produce "unintuitive" (but
correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial.laguerre import lagmul
>>> lagmul([1, 2, 3], [0, 1, 2])
array([ 8., -13., 38., -51., 36.])
"""
# s1, s2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if len(c1) > len(c2):
c = c2
xs = c1
else:
c = c1
xs = c2
if len(c) == 1:
c0 = c[0]*xs
c1 = 0
elif len(c) == 2:
c0 = c[0]*xs
c1 = c[1]*xs
else:
nd = len(c)
c0 = c[-2]*xs
c1 = c[-1]*xs
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = lagsub(c[-i]*xs, (c1*(nd - 1))/nd)
c1 = lagadd(tmp, lagsub((2*nd - 1)*c1, lagmulx(c1))/nd)
return lagadd(c0, lagsub(c1, lagmulx(c1)))
The provided code snippet includes necessary dependencies for implementing the `lagpow` function. Write a Python function `def lagpow(c, pow, maxpower=16)` to solve the following problem:
Raise a Laguerre series to a power. Returns the Laguerre series `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` Parameters ---------- c : array_like 1-D array of Laguerre series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Laguerre series of power. See Also -------- lagadd, lagsub, lagmulx, lagmul, lagdiv Examples -------- >>> from numpy.polynomial.laguerre import lagpow >>> lagpow([1, 2, 3], 2) array([ 14., -16., 56., -72., 54.])
Here is the function:
def lagpow(c, pow, maxpower=16):
"""Raise a Laguerre series to a power.
Returns the Laguerre series `c` raised to the power `pow`. The
argument `c` is a sequence of coefficients ordered from low to high.
i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.``
Parameters
----------
c : array_like
1-D array of Laguerre series coefficients ordered from low to
high.
pow : integer
Power to which the series will be raised
maxpower : integer, optional
Maximum power allowed. This is mainly to limit growth of the series
to unmanageable size. Default is 16
Returns
-------
coef : ndarray
Laguerre series of power.
See Also
--------
lagadd, lagsub, lagmulx, lagmul, lagdiv
Examples
--------
>>> from numpy.polynomial.laguerre import lagpow
>>> lagpow([1, 2, 3], 2)
array([ 14., -16., 56., -72., 54.])
"""
return pu._pow(lagmul, c, pow, maxpower) | Raise a Laguerre series to a power. Returns the Laguerre series `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` Parameters ---------- c : array_like 1-D array of Laguerre series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Laguerre series of power. See Also -------- lagadd, lagsub, lagmulx, lagmul, lagdiv Examples -------- >>> from numpy.polynomial.laguerre import lagpow >>> lagpow([1, 2, 3], 2) array([ 14., -16., 56., -72., 54.]) |
169,764 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagval(x, c, tensor=True):
"""
Evaluate a Laguerre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
lagval2d, laggrid2d, lagval3d, laggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.laguerre import lagval
>>> coef = [1,2,3]
>>> lagval(1, coef)
-0.5
>>> lagval([[1,2],[3,4]], coef)
array([[-0.5, -4. ],
[-4.5, -2. ]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*((2*nd - 1) - x))/nd
return c0 + c1*(1 - x)
The provided code snippet includes necessary dependencies for implementing the `lagint` function. Write a Python function `def lagint(c, m=1, k=[], lbnd=0, scl=1, axis=0)` to solve the following problem:
Integrate a Laguerre series. Returns the Laguerre series coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Laguerre series coefficients. If `c` is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at ``lbnd`` is the first value in the list, the value of the second integral at ``lbnd`` is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray Laguerre series coefficients of the integral. Raises ------ ValueError If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- lagder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a` - perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial.laguerre import lagint >>> lagint([1,2,3]) array([ 1., 1., 1., -3.]) >>> lagint([1,2,3], m=2) array([ 1., 0., 0., -4., 3.]) >>> lagint([1,2,3], k=1) array([ 2., 1., 1., -3.]) >>> lagint([1,2,3], lbnd=-1) array([11.5, 1. , 1. , -3. ]) >>> lagint([1,2], m=2, k=[1,2], lbnd=-1) array([ 11.16666667, -5. , -3. , 2. ]) # may vary
Here is the function:
def lagint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
"""
Integrate a Laguerre series.
Returns the Laguerre series coefficients `c` integrated `m` times from
`lbnd` along `axis`. At each iteration the resulting series is
**multiplied** by `scl` and an integration constant, `k`, is added.
The scaling factor is for use in a linear change of variable. ("Buyer
beware": note that, depending on what one is doing, one may want `scl`
to be the reciprocal of what one might expect; for more information,
see the Notes section below.) The argument `c` is an array of
coefficients from low to high degree along each axis, e.g., [1,2,3]
represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
Parameters
----------
c : array_like
Array of Laguerre series coefficients. If `c` is multidimensional
the different axis correspond to different variables with the
degree in each axis given by the corresponding index.
m : int, optional
Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
Integration constant(s). The value of the first integral at
``lbnd`` is the first value in the list, the value of the second
integral at ``lbnd`` is the second value, etc. If ``k == []`` (the
default), all constants are set to zero. If ``m == 1``, a single
scalar can be given instead of a list.
lbnd : scalar, optional
The lower bound of the integral. (Default: 0)
scl : scalar, optional
Following each integration the result is *multiplied* by `scl`
before the integration constant is added. (Default: 1)
axis : int, optional
Axis over which the integral is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
S : ndarray
Laguerre series coefficients of the integral.
Raises
------
ValueError
If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
``np.ndim(scl) != 0``.
See Also
--------
lagder
Notes
-----
Note that the result of each integration is *multiplied* by `scl`.
Why is this important to note? Say one is making a linear change of
variable :math:`u = ax + b` in an integral relative to `x`. Then
:math:`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a` - perhaps not what one would have first thought.
Also note that, in general, the result of integrating a C-series needs
to be "reprojected" onto the C-series basis set. Thus, typically,
the result of this function is "unintuitive," albeit correct; see
Examples section below.
Examples
--------
>>> from numpy.polynomial.laguerre import lagint
>>> lagint([1,2,3])
array([ 1., 1., 1., -3.])
>>> lagint([1,2,3], m=2)
array([ 1., 0., 0., -4., 3.])
>>> lagint([1,2,3], k=1)
array([ 2., 1., 1., -3.])
>>> lagint([1,2,3], lbnd=-1)
array([11.5, 1. , 1. , -3. ])
>>> lagint([1,2], m=2, k=[1,2], lbnd=-1)
array([ 11.16666667, -5. , -3. , 2. ]) # may vary
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if not np.iterable(k):
k = [k]
cnt = pu._deprecate_as_int(m, "the order of integration")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of integration must be non-negative")
if len(k) > cnt:
raise ValueError("Too many integration constants")
if np.ndim(lbnd) != 0:
raise ValueError("lbnd must be a scalar.")
if np.ndim(scl) != 0:
raise ValueError("scl must be a scalar.")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
k = list(k) + [0]*(cnt - len(k))
for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
c[0] += k[i]
else:
tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
tmp[0] = c[0]
tmp[1] = -c[0]
for j in range(1, n):
tmp[j] += c[j]
tmp[j + 1] = -c[j]
tmp[0] += k[i] - lagval(lbnd, tmp)
c = tmp
c = np.moveaxis(c, 0, iaxis)
return c | Integrate a Laguerre series. Returns the Laguerre series coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Laguerre series coefficients. If `c` is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at ``lbnd`` is the first value in the list, the value of the second integral at ``lbnd`` is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray Laguerre series coefficients of the integral. Raises ------ ValueError If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- lagder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a` - perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial.laguerre import lagint >>> lagint([1,2,3]) array([ 1., 1., 1., -3.]) >>> lagint([1,2,3], m=2) array([ 1., 0., 0., -4., 3.]) >>> lagint([1,2,3], k=1) array([ 2., 1., 1., -3.]) >>> lagint([1,2,3], lbnd=-1) array([11.5, 1. , 1. , -3. ]) >>> lagint([1,2], m=2, k=[1,2], lbnd=-1) array([ 11.16666667, -5. , -3. , 2. ]) # may vary |
169,765 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagval(x, c, tensor=True):
"""
Evaluate a Laguerre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
lagval2d, laggrid2d, lagval3d, laggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.laguerre import lagval
>>> coef = [1,2,3]
>>> lagval(1, coef)
-0.5
>>> lagval([[1,2],[3,4]], coef)
array([[-0.5, -4. ],
[-4.5, -2. ]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*((2*nd - 1) - x))/nd
return c0 + c1*(1 - x)
The provided code snippet includes necessary dependencies for implementing the `lagval2d` function. Write a Python function `def lagval2d(x, y, c)` to solve the following problem:
Evaluate a 2-D Laguerre series at points (x, y). This function returns the values: .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y) The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points formed with pairs of corresponding values from `x` and `y`. See Also -------- lagval, laggrid2d, lagval3d, laggrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def lagval2d(x, y, c):
"""
Evaluate a 2-D Laguerre series at points (x, y).
This function returns the values:
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars and they
must have the same shape after conversion. In either case, either `x`
and `y` or their elements must support multiplication and addition both
with themselves and with the elements of `c`.
If `c` is a 1-D array a one is implicitly appended to its shape to make
it 2-D. The shape of the result will be c.shape[2:] + x.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points `(x, y)`,
where `x` and `y` must have the same shape. If `x` or `y` is a list
or tuple, it is first converted to an ndarray, otherwise it is left
unchanged and if it isn't an ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term
of multi-degree i,j is contained in ``c[i,j]``. If `c` has
dimension greater than two the remaining indices enumerate multiple
sets of coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points formed with
pairs of corresponding values from `x` and `y`.
See Also
--------
lagval, laggrid2d, lagval3d, laggrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(lagval, c, x, y) | Evaluate a 2-D Laguerre series at points (x, y). This function returns the values: .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y) The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points formed with pairs of corresponding values from `x` and `y`. See Also -------- lagval, laggrid2d, lagval3d, laggrid3d Notes ----- .. versionadded:: 1.7.0 |
169,766 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagval(x, c, tensor=True):
"""
Evaluate a Laguerre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
lagval2d, laggrid2d, lagval3d, laggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.laguerre import lagval
>>> coef = [1,2,3]
>>> lagval(1, coef)
-0.5
>>> lagval([[1,2],[3,4]], coef)
array([[-0.5, -4. ],
[-4.5, -2. ]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*((2*nd - 1) - x))/nd
return c0 + c1*(1 - x)
The provided code snippet includes necessary dependencies for implementing the `laggrid2d` function. Write a Python function `def laggrid2d(x, y, c)` to solve the following problem:
Evaluate a 2-D Laguerre series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b) where the points `(a, b)` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Chebyshev series at points in the Cartesian product of `x` and `y`. See Also -------- lagval, lagval2d, lagval3d, laggrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def laggrid2d(x, y, c):
"""
Evaluate a 2-D Laguerre series on the Cartesian product of x and y.
This function returns the values:
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)
where the points `(a, b)` consist of all pairs formed by taking
`a` from `x` and `b` from `y`. The resulting points form a grid with
`x` in the first dimension and `y` in the second.
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars. In either
case, either `x` and `y` or their elements must support multiplication
and addition both with themselves and with the elements of `c`.
If `c` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape + y.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points in the
Cartesian product of `x` and `y`. If `x` or `y` is a list or
tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional Chebyshev series at points in the
Cartesian product of `x` and `y`.
See Also
--------
lagval, lagval2d, lagval3d, laggrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(lagval, c, x, y) | Evaluate a 2-D Laguerre series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b) where the points `(a, b)` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Chebyshev series at points in the Cartesian product of `x` and `y`. See Also -------- lagval, lagval2d, lagval3d, laggrid3d Notes ----- .. versionadded:: 1.7.0 |
169,767 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagval(x, c, tensor=True):
"""
Evaluate a Laguerre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
lagval2d, laggrid2d, lagval3d, laggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.laguerre import lagval
>>> coef = [1,2,3]
>>> lagval(1, coef)
-0.5
>>> lagval([[1,2],[3,4]], coef)
array([[-0.5, -4. ],
[-4.5, -2. ]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*((2*nd - 1) - x))/nd
return c0 + c1*(1 - x)
The provided code snippet includes necessary dependencies for implementing the `lagval3d` function. Write a Python function `def lagval3d(x, y, z, c)` to solve the following problem:
Evaluate a 3-D Laguerre series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z) The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- lagval, lagval2d, laggrid2d, laggrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def lagval3d(x, y, z, c):
"""
Evaluate a 3-D Laguerre series at points (x, y, z).
This function returns the values:
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)
The parameters `x`, `y`, and `z` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
`x`, `y`, and `z` or their elements must support multiplication and
addition both with themselves and with the elements of `c`.
If `c` has fewer than 3 dimensions, ones are implicitly appended to its
shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape.
Parameters
----------
x, y, z : array_like, compatible object
The three dimensional series is evaluated at the points
`(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If
any of `x`, `y`, or `z` is a list or tuple, it is first converted
to an ndarray, otherwise it is left unchanged and if it isn't an
ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
greater than 3 the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the multidimensional polynomial on points formed with
triples of corresponding values from `x`, `y`, and `z`.
See Also
--------
lagval, lagval2d, laggrid2d, laggrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(lagval, c, x, y, z) | Evaluate a 3-D Laguerre series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z) The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- lagval, lagval2d, laggrid2d, laggrid3d Notes ----- .. versionadded:: 1.7.0 |
169,768 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagval(x, c, tensor=True):
"""
Evaluate a Laguerre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
lagval2d, laggrid2d, lagval3d, laggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.laguerre import lagval
>>> coef = [1,2,3]
>>> lagval(1, coef)
-0.5
>>> lagval([[1,2],[3,4]], coef)
array([[-0.5, -4. ],
[-4.5, -2. ]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*((2*nd - 1) - x))/nd
return c0 + c1*(1 - x)
The provided code snippet includes necessary dependencies for implementing the `laggrid3d` function. Write a Python function `def laggrid3d(x, y, z, c)` to solve the following problem:
Evaluate a 3-D Laguerre series on the Cartesian product of x, y, and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c) where the points `(a, b, c)` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- lagval, lagval2d, laggrid2d, lagval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def laggrid3d(x, y, z, c):
"""
Evaluate a 3-D Laguerre series on the Cartesian product of x, y, and z.
This function returns the values:
.. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c)
where the points `(a, b, c)` consist of all triples formed by taking
`a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
a grid with `x` in the first dimension, `y` in the second, and `z` in
the third.
The parameters `x`, `y`, and `z` are converted to arrays only if they
are tuples or a lists, otherwise they are treated as a scalars. In
either case, either `x`, `y`, and `z` or their elements must support
multiplication and addition both with themselves and with the elements
of `c`.
If `c` has fewer than three dimensions, ones are implicitly appended to
its shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape + y.shape + z.shape.
Parameters
----------
x, y, z : array_like, compatible objects
The three dimensional series is evaluated at the points in the
Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a
list or tuple, it is first converted to an ndarray, otherwise it is
left unchanged and, if it isn't an ndarray, it is treated as a
scalar.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree i,j are contained in ``c[i,j]``. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points in the Cartesian
product of `x` and `y`.
See Also
--------
lagval, lagval2d, laggrid2d, lagval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(lagval, c, x, y, z) | Evaluate a 3-D Laguerre series on the Cartesian product of x, y, and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c) where the points `(a, b, c)` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- lagval, lagval2d, laggrid2d, lagval3d Notes ----- .. versionadded:: 1.7.0 |
169,769 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = L_i(x)
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Laguerre polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
array ``V = lagvander(x, n)``, then ``np.dot(V, c)`` and
``lagval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Laguerre series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo-Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Laguerre polynomial. The dtype will be the same as
the converted `x`.
Examples
--------
>>> from numpy.polynomial.laguerre import lagvander
>>> x = np.array([0, 1, 2])
>>> lagvander(x, 3)
array([[ 1. , 1. , 1. , 1. ],
[ 1. , 0. , -0.5 , -0.66666667],
[ 1. , -1. , -1. , -0.33333333]])
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
if ideg > 0:
v[1] = 1 - x
for i in range(2, ideg + 1):
v[i] = (v[i-1]*(2*i - 1 - x) - v[i-2]*(i - 1))/i
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `lagvander2d` function. Write a Python function `def lagvander2d(x, y, deg)` to solve the following problem:
Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y)`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y), where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of `V` index the points `(x, y)` and the last index encodes the degrees of the Laguerre polynomials. If ``V = lagvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``lagval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Laguerre series of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- lagvander, lagvander3d, lagval2d, lagval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def lagvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y)`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y),
where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
`V` index the points `(x, y)` and the last index encodes the degrees of
the Laguerre polynomials.
If ``V = lagvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
correspond to the elements of a 2-D coefficient array `c` of shape
(xdeg + 1, ydeg + 1) in the order
.. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
and ``np.dot(V, c.flat)`` and ``lagval2d(x, y, c)`` will be the same
up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 2-D Laguerre
series of the same degrees and sample points.
Parameters
----------
x, y : array_like
Arrays of point coordinates, all of the same shape. The dtypes
will be converted to either float64 or complex128 depending on
whether any of the elements are complex. Scalars are converted to
1-D arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg].
Returns
-------
vander2d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same
as the converted `x` and `y`.
See Also
--------
lagvander, lagvander3d, lagval2d, lagval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((lagvander, lagvander), (x, y), deg) | Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y)`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y), where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of `V` index the points `(x, y)` and the last index encodes the degrees of the Laguerre polynomials. If ``V = lagvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``lagval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Laguerre series of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- lagvander, lagvander3d, lagval2d, lagval3d Notes ----- .. versionadded:: 1.7.0 |
169,770 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = L_i(x)
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Laguerre polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
array ``V = lagvander(x, n)``, then ``np.dot(V, c)`` and
``lagval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Laguerre series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo-Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Laguerre polynomial. The dtype will be the same as
the converted `x`.
Examples
--------
>>> from numpy.polynomial.laguerre import lagvander
>>> x = np.array([0, 1, 2])
>>> lagvander(x, 3)
array([[ 1. , 1. , 1. , 1. ],
[ 1. , 0. , -0.5 , -0.66666667],
[ 1. , -1. , -1. , -0.33333333]])
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
if ideg > 0:
v[1] = 1 - x
for i in range(2, ideg + 1):
v[i] = (v[i-1]*(2*i - 1 - x) - v[i-2]*(i - 1))/i
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `lagvander3d` function. Write a Python function `def lagvander3d(x, y, z, deg)` to solve the following problem:
Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z), where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading indices of `V` index the points `(x, y, z)` and the last index encodes the degrees of the Laguerre polynomials. If ``V = lagvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``lagval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Laguerre series of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- lagvander, lagvander3d, lagval2d, lagval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def lagvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
then The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),
where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading
indices of `V` index the points `(x, y, z)` and the last index encodes
the degrees of the Laguerre polynomials.
If ``V = lagvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
of `V` correspond to the elements of a 3-D coefficient array `c` of
shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
.. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
and ``np.dot(V, c.flat)`` and ``lagval3d(x, y, z, c)`` will be the
same up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 3-D Laguerre
series of the same degrees and sample points.
Parameters
----------
x, y, z : array_like
Arrays of point coordinates, all of the same shape. The dtypes will
be converted to either float64 or complex128 depending on whether
any of the elements are complex. Scalars are converted to 1-D
arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg, z_deg].
Returns
-------
vander3d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will
be the same as the converted `x`, `y`, and `z`.
See Also
--------
lagvander, lagvander3d, lagval2d, lagval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((lagvander, lagvander, lagvander), (x, y, z), deg) | Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z), where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading indices of `V` index the points `(x, y, z)` and the last index encodes the degrees of the Laguerre polynomials. If ``V = lagvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``lagval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Laguerre series of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- lagvander, lagvander3d, lagval2d, lagval3d Notes ----- .. versionadded:: 1.7.0 |
169,771 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = L_i(x)
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Laguerre polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
array ``V = lagvander(x, n)``, then ``np.dot(V, c)`` and
``lagval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Laguerre series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo-Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Laguerre polynomial. The dtype will be the same as
the converted `x`.
Examples
--------
>>> from numpy.polynomial.laguerre import lagvander
>>> x = np.array([0, 1, 2])
>>> lagvander(x, 3)
array([[ 1. , 1. , 1. , 1. ],
[ 1. , 0. , -0.5 , -0.66666667],
[ 1. , -1. , -1. , -0.33333333]])
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
if ideg > 0:
v[1] = 1 - x
for i in range(2, ideg + 1):
v[i] = (v[i-1]*(2*i - 1 - x) - v[i-2]*(i - 1))/i
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `lagfit` function. Write a Python function `def lagfit(x, y, deg, rcond=None, full=False, w=None)` to solve the following problem:
Least squares fit of Laguerre series to data. Return the coefficients of a Laguerre series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x), where ``n`` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the weight ``w[i]`` applies to the unsquared residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. When using inverse-variance weighting, use ``w[i] = 1/sigma(y[i])``. The default value is None. Returns ------- coef : ndarray, shape (M,) or (M, K) Laguerre coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column *k* of `y` are in column *k*. [residuals, rank, singular_values, rcond] : list These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the numerical rank of the scaled Vandermonde matrix - singular_values -- singular values of the scaled Vandermonde matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if ``full == False``. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- numpy.polynomial.polynomial.polyfit numpy.polynomial.legendre.legfit numpy.polynomial.chebyshev.chebfit numpy.polynomial.hermite.hermfit numpy.polynomial.hermite_e.hermefit lagval : Evaluates a Laguerre series. lagvander : pseudo Vandermonde matrix of Laguerre series. lagweight : Laguerre weight function. numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Laguerre series ``p`` that minimizes the sum of the weighted squared errors .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, where the :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where ``V`` is the weighted pseudo Vandermonde matrix of `x`, ``c`` are the coefficients to be solved for, `w` are the weights, and `y` are the observed values. This equation is then solved using the singular value decomposition of ``V``. If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Laguerre series are probably most useful when the data can be approximated by ``sqrt(w(x)) * p(x)``, where ``w(x)`` is the Laguerre weight. In that case the weight ``sqrt(w(x[i]))`` should be used together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is available as `lagweight`. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples -------- >>> from numpy.polynomial.laguerre import lagfit, lagval >>> x = np.linspace(0, 10) >>> err = np.random.randn(len(x))/10 >>> y = lagval(x, [1, 2, 3]) + err >>> lagfit(x, y, 2) array([ 0.96971004, 2.00193749, 3.00288744]) # may vary
Here is the function:
def lagfit(x, y, deg, rcond=None, full=False, w=None):
"""
Least squares fit of Laguerre series to data.
Return the coefficients of a Laguerre series of degree `deg` that is the
least squares fit to the data values `y` given at points `x`. If `y` is
1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
fits are done, one for each column of `y`, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form
.. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),
where ``n`` is `deg`.
Parameters
----------
x : array_like, shape (M,)
x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,) or (M, K)
y-coordinates of the sample points. Several data sets of sample
points sharing the same x-coordinates can be fitted at once by
passing in a 2D-array that contains one dataset per column.
deg : int or 1-D array_like
Degree(s) of the fitting polynomials. If `deg` is a single integer
all terms up to and including the `deg`'th term are included in the
fit. For NumPy versions >= 1.11.0 a list of integers specifying the
degrees of the terms to include may be used instead.
rcond : float, optional
Relative condition number of the fit. Singular values smaller than
this relative to the largest singular value will be ignored. The
default value is len(x)*eps, where eps is the relative precision of
the float type, about 2e-16 in most cases.
full : bool, optional
Switch determining nature of return value. When it is False (the
default) just the coefficients are returned, when True diagnostic
information from the singular value decomposition is also returned.
w : array_like, shape (`M`,), optional
Weights. If not None, the weight ``w[i]`` applies to the unsquared
residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
chosen so that the errors of the products ``w[i]*y[i]`` all have the
same variance. When using inverse-variance weighting, use
``w[i] = 1/sigma(y[i])``. The default value is None.
Returns
-------
coef : ndarray, shape (M,) or (M, K)
Laguerre coefficients ordered from low to high. If `y` was 2-D,
the coefficients for the data in column *k* of `y` are in column
*k*.
[residuals, rank, singular_values, rcond] : list
These values are only returned if ``full == True``
- residuals -- sum of squared residuals of the least squares fit
- rank -- the numerical rank of the scaled Vandermonde matrix
- singular_values -- singular values of the scaled Vandermonde matrix
- rcond -- value of `rcond`.
For more details, see `numpy.linalg.lstsq`.
Warns
-----
RankWarning
The rank of the coefficient matrix in the least-squares fit is
deficient. The warning is only raised if ``full == False``. The
warnings can be turned off by
>>> import warnings
>>> warnings.simplefilter('ignore', np.RankWarning)
See Also
--------
numpy.polynomial.polynomial.polyfit
numpy.polynomial.legendre.legfit
numpy.polynomial.chebyshev.chebfit
numpy.polynomial.hermite.hermfit
numpy.polynomial.hermite_e.hermefit
lagval : Evaluates a Laguerre series.
lagvander : pseudo Vandermonde matrix of Laguerre series.
lagweight : Laguerre weight function.
numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes
-----
The solution is the coefficients of the Laguerre series ``p`` that
minimizes the sum of the weighted squared errors
.. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
where the :math:`w_j` are the weights. This problem is solved by
setting up as the (typically) overdetermined matrix equation
.. math:: V(x) * c = w * y,
where ``V`` is the weighted pseudo Vandermonde matrix of `x`, ``c`` are the
coefficients to be solved for, `w` are the weights, and `y` are the
observed values. This equation is then solved using the singular value
decomposition of ``V``.
If some of the singular values of `V` are so small that they are
neglected, then a `RankWarning` will be issued. This means that the
coefficient values may be poorly determined. Using a lower order fit
will usually get rid of the warning. The `rcond` parameter can also be
set to a value smaller than its default, but the resulting fit may be
spurious and have large contributions from roundoff error.
Fits using Laguerre series are probably most useful when the data can
be approximated by ``sqrt(w(x)) * p(x)``, where ``w(x)`` is the Laguerre
weight. In that case the weight ``sqrt(w(x[i]))`` should be used
together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is
available as `lagweight`.
References
----------
.. [1] Wikipedia, "Curve fitting",
https://en.wikipedia.org/wiki/Curve_fitting
Examples
--------
>>> from numpy.polynomial.laguerre import lagfit, lagval
>>> x = np.linspace(0, 10)
>>> err = np.random.randn(len(x))/10
>>> y = lagval(x, [1, 2, 3]) + err
>>> lagfit(x, y, 2)
array([ 0.96971004, 2.00193749, 3.00288744]) # may vary
"""
return pu._fit(lagvander, x, y, deg, rcond, full, w) | Least squares fit of Laguerre series to data. Return the coefficients of a Laguerre series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x), where ``n`` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the weight ``w[i]`` applies to the unsquared residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. When using inverse-variance weighting, use ``w[i] = 1/sigma(y[i])``. The default value is None. Returns ------- coef : ndarray, shape (M,) or (M, K) Laguerre coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column *k* of `y` are in column *k*. [residuals, rank, singular_values, rcond] : list These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the numerical rank of the scaled Vandermonde matrix - singular_values -- singular values of the scaled Vandermonde matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if ``full == False``. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- numpy.polynomial.polynomial.polyfit numpy.polynomial.legendre.legfit numpy.polynomial.chebyshev.chebfit numpy.polynomial.hermite.hermfit numpy.polynomial.hermite_e.hermefit lagval : Evaluates a Laguerre series. lagvander : pseudo Vandermonde matrix of Laguerre series. lagweight : Laguerre weight function. numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Laguerre series ``p`` that minimizes the sum of the weighted squared errors .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, where the :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where ``V`` is the weighted pseudo Vandermonde matrix of `x`, ``c`` are the coefficients to be solved for, `w` are the weights, and `y` are the observed values. This equation is then solved using the singular value decomposition of ``V``. If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Laguerre series are probably most useful when the data can be approximated by ``sqrt(w(x)) * p(x)``, where ``w(x)`` is the Laguerre weight. In that case the weight ``sqrt(w(x[i]))`` should be used together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is available as `lagweight`. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples -------- >>> from numpy.polynomial.laguerre import lagfit, lagval >>> x = np.linspace(0, 10) >>> err = np.random.randn(len(x))/10 >>> y = lagval(x, [1, 2, 3]) + err >>> lagfit(x, y, 2) array([ 0.96971004, 2.00193749, 3.00288744]) # may vary |
169,772 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagcompanion(c):
"""
Return the companion matrix of c.
The usual companion matrix of the Laguerre polynomials is already
symmetric when `c` is a basis Laguerre polynomial, so no scaling is
applied.
Parameters
----------
c : array_like
1-D array of Laguerre series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded:: 1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[1 + c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
top = mat.reshape(-1)[1::n+1]
mid = mat.reshape(-1)[0::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = -np.arange(1, n)
mid[...] = 2.*np.arange(n) + 1.
bot[...] = top
mat[:, -1] += (c[:-1]/c[-1])*n
return mat
The provided code snippet includes necessary dependencies for implementing the `lagroots` function. Write a Python function `def lagroots(c)` to solve the following problem:
Compute the roots of a Laguerre series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * L_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex. See Also -------- numpy.polynomial.polynomial.polyroots numpy.polynomial.legendre.legroots numpy.polynomial.chebyshev.chebroots numpy.polynomial.hermite.hermroots numpy.polynomial.hermite_e.hermeroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method. The Laguerre series basis polynomials aren't powers of `x` so the results of this function may seem unintuitive. Examples -------- >>> from numpy.polynomial.laguerre import lagroots, lagfromroots >>> coef = lagfromroots([0, 1, 2]) >>> coef array([ 2., -8., 12., -6.]) >>> lagroots(coef) array([-4.4408921e-16, 1.0000000e+00, 2.0000000e+00])
Here is the function:
def lagroots(c):
"""
Compute the roots of a Laguerre series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * L_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
numpy.polynomial.polynomial.polyroots
numpy.polynomial.legendre.legroots
numpy.polynomial.chebyshev.chebroots
numpy.polynomial.hermite.hermroots
numpy.polynomial.hermite_e.hermeroots
Notes
-----
The root estimates are obtained as the eigenvalues of the companion
matrix, Roots far from the origin of the complex plane may have large
errors due to the numerical instability of the series for such
values. Roots with multiplicity greater than 1 will also show larger
errors as the value of the series near such points is relatively
insensitive to errors in the roots. Isolated roots near the origin can
be improved by a few iterations of Newton's method.
The Laguerre series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> from numpy.polynomial.laguerre import lagroots, lagfromroots
>>> coef = lagfromroots([0, 1, 2])
>>> coef
array([ 2., -8., 12., -6.])
>>> lagroots(coef)
array([-4.4408921e-16, 1.0000000e+00, 2.0000000e+00])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) <= 1:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([1 + c[0]/c[1]])
# rotated companion matrix reduces error
m = lagcompanion(c)[::-1,::-1]
r = la.eigvals(m)
r.sort()
return r | Compute the roots of a Laguerre series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * L_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex. See Also -------- numpy.polynomial.polynomial.polyroots numpy.polynomial.legendre.legroots numpy.polynomial.chebyshev.chebroots numpy.polynomial.hermite.hermroots numpy.polynomial.hermite_e.hermeroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method. The Laguerre series basis polynomials aren't powers of `x` so the results of this function may seem unintuitive. Examples -------- >>> from numpy.polynomial.laguerre import lagroots, lagfromroots >>> coef = lagfromroots([0, 1, 2]) >>> coef array([ 2., -8., 12., -6.]) >>> lagroots(coef) array([-4.4408921e-16, 1.0000000e+00, 2.0000000e+00]) |
169,773 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def lagder(c, m=1, scl=1, axis=0):
"""
Differentiate a Laguerre series.
Returns the Laguerre series coefficients `c` differentiated `m` times
along `axis`. At each iteration the result is multiplied by `scl` (the
scaling factor is for use in a linear change of variable). The argument
`c` is an array of coefficients from low to high degree along each
axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2``
while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
``y``.
Parameters
----------
c : array_like
Array of Laguerre series coefficients. If `c` is multidimensional
the different axis correspond to different variables with the
degree in each axis given by the corresponding index.
m : int, optional
Number of derivatives taken, must be non-negative. (Default: 1)
scl : scalar, optional
Each differentiation is multiplied by `scl`. The end result is
multiplication by ``scl**m``. This is for use in a linear change of
variable. (Default: 1)
axis : int, optional
Axis over which the derivative is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
der : ndarray
Laguerre series of the derivative.
See Also
--------
lagint
Notes
-----
In general, the result of differentiating a Laguerre series does not
resemble the same operation on a power series. Thus the result of this
function may be "unintuitive," albeit correct; see Examples section
below.
Examples
--------
>>> from numpy.polynomial.laguerre import lagder
>>> lagder([ 1., 1., 1., -3.])
array([1., 2., 3.])
>>> lagder([ 1., 0., 0., -4., 3.], m=2)
array([1., 2., 3.])
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
cnt = pu._deprecate_as_int(m, "the order of derivation")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of derivation must be non-negative")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
n = len(c)
if cnt >= n:
c = c[:1]*0
else:
for i in range(cnt):
n = n - 1
c *= scl
der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
for j in range(n, 1, -1):
der[j - 1] = -c[j]
c[j - 1] += c[j]
der[0] = -c[1]
c = der
c = np.moveaxis(c, 0, iaxis)
return c
def lagval(x, c, tensor=True):
"""
Evaluate a Laguerre series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
lagval2d, laggrid2d, lagval3d, laggrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
Examples
--------
>>> from numpy.polynomial.laguerre import lagval
>>> coef = [1,2,3]
>>> lagval(1, coef)
-0.5
>>> lagval([[1,2],[3,4]], coef)
array([[-0.5, -4. ],
[-4.5, -2. ]])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
nd = len(c)
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
nd = nd - 1
c0 = c[-i] - (c1*(nd - 1))/nd
c1 = tmp + (c1*((2*nd - 1) - x))/nd
return c0 + c1*(1 - x)
def lagcompanion(c):
"""
Return the companion matrix of c.
The usual companion matrix of the Laguerre polynomials is already
symmetric when `c` is a basis Laguerre polynomial, so no scaling is
applied.
Parameters
----------
c : array_like
1-D array of Laguerre series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded:: 1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[1 + c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
top = mat.reshape(-1)[1::n+1]
mid = mat.reshape(-1)[0::n+1]
bot = mat.reshape(-1)[n::n+1]
top[...] = -np.arange(1, n)
mid[...] = 2.*np.arange(n) + 1.
bot[...] = top
mat[:, -1] += (c[:-1]/c[-1])*n
return mat
The provided code snippet includes necessary dependencies for implementing the `laggauss` function. Write a Python function `def laggauss(deg)` to solve the following problem:
Gauss-Laguerre quadrature. Computes the sample points and weights for Gauss-Laguerre quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`[0, \\inf]` with the weight function :math:`f(x) = \\exp(-x)`. Parameters ---------- deg : int Number of sample points and weights. It must be >= 1. Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights. Notes ----- .. versionadded:: 1.7.0 The results have only been tested up to degree 100 higher degrees may be problematic. The weights are determined by using the fact that .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k)) where :math:`c` is a constant independent of :math:`k` and :math:`x_k` is the k'th root of :math:`L_n`, and then scaling the results to get the right value when integrating 1.
Here is the function:
def laggauss(deg):
"""
Gauss-Laguerre quadrature.
Computes the sample points and weights for Gauss-Laguerre quadrature.
These sample points and weights will correctly integrate polynomials of
degree :math:`2*deg - 1` or less over the interval :math:`[0, \\inf]`
with the weight function :math:`f(x) = \\exp(-x)`.
Parameters
----------
deg : int
Number of sample points and weights. It must be >= 1.
Returns
-------
x : ndarray
1-D ndarray containing the sample points.
y : ndarray
1-D ndarray containing the weights.
Notes
-----
.. versionadded:: 1.7.0
The results have only been tested up to degree 100 higher degrees may
be problematic. The weights are determined by using the fact that
.. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k))
where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
is the k'th root of :math:`L_n`, and then scaling the results to get
the right value when integrating 1.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg <= 0:
raise ValueError("deg must be a positive integer")
# first approximation of roots. We use the fact that the companion
# matrix is symmetric in this case in order to obtain better zeros.
c = np.array([0]*deg + [1])
m = lagcompanion(c)
x = la.eigvalsh(m)
# improve roots by one application of Newton
dy = lagval(x, c)
df = lagval(x, lagder(c))
x -= dy/df
# compute the weights. We scale the factor to avoid possible numerical
# overflow.
fm = lagval(x, c[1:])
fm /= np.abs(fm).max()
df /= np.abs(df).max()
w = 1/(fm * df)
# scale w to get the right value, 1 in this case
w /= w.sum()
return x, w | Gauss-Laguerre quadrature. Computes the sample points and weights for Gauss-Laguerre quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`[0, \\inf]` with the weight function :math:`f(x) = \\exp(-x)`. Parameters ---------- deg : int Number of sample points and weights. It must be >= 1. Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights. Notes ----- .. versionadded:: 1.7.0 The results have only been tested up to degree 100 higher degrees may be problematic. The weights are determined by using the fact that .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k)) where :math:`c` is a constant independent of :math:`k` and :math:`x_k` is the k'th root of :math:`L_n`, and then scaling the results to get the right value when integrating 1. |
169,774 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `lagweight` function. Write a Python function `def lagweight(x)` to solve the following problem:
Weight function of the Laguerre polynomials. The weight function is :math:`exp(-x)` and the interval of integration is :math:`[0, \\inf]`. The Laguerre polynomials are orthogonal, but not normalized, with respect to this weight function. Parameters ---------- x : array_like Values at which the weight function will be computed. Returns ------- w : ndarray The weight function at `x`. Notes ----- .. versionadded:: 1.7.0
Here is the function:
def lagweight(x):
"""Weight function of the Laguerre polynomials.
The weight function is :math:`exp(-x)` and the interval of integration
is :math:`[0, \\inf]`. The Laguerre polynomials are orthogonal, but not
normalized, with respect to this weight function.
Parameters
----------
x : array_like
Values at which the weight function will be computed.
Returns
-------
w : ndarray
The weight function at `x`.
Notes
-----
.. versionadded:: 1.7.0
"""
w = np.exp(-x)
return w | Weight function of the Laguerre polynomials. The weight function is :math:`exp(-x)` and the interval of integration is :math:`[0, \\inf]`. The Laguerre polynomials are orthogonal, but not normalized, with respect to this weight function. Parameters ---------- x : array_like Values at which the weight function will be computed. Returns ------- w : ndarray The weight function at `x`. Notes ----- .. versionadded:: 1.7.0 |
169,775 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def _zseries_div(z1, z2):
"""Divide the first z-series by the second.
Divide `z1` by `z2` and return the quotient and remainder as z-series.
Warning: this implementation only applies when both z1 and z2 have the
same symmetry, which is sufficient for present purposes.
Parameters
----------
z1, z2 : 1-D ndarray
The arrays must be 1-D and have the same symmetry, but this is not
checked.
Returns
-------
(quotient, remainder) : 1-D ndarrays
Quotient and remainder as z-series.
Notes
-----
This is not the same as polynomial division on account of the desired form
of the remainder. If symmetric/anti-symmetric z-series are denoted by S/A
then the following rules apply:
S/S -> S,S
A/A -> S,A
The restriction to types of the same symmetry could be fixed but seems like
unneeded generality. There is no natural form for the remainder in the case
where there is no symmetry.
"""
z1 = z1.copy()
z2 = z2.copy()
lc1 = len(z1)
lc2 = len(z2)
if lc2 == 1:
z1 /= z2
return z1, z1[:1]*0
elif lc1 < lc2:
return z1[:1]*0, z1
else:
dlen = lc1 - lc2
scl = z2[0]
z2 /= scl
quo = np.empty(dlen + 1, dtype=z1.dtype)
i = 0
j = dlen
while i < j:
r = z1[i]
quo[i] = z1[i]
quo[dlen - i] = r
tmp = r*z2
z1[i:i+lc2] -= tmp
z1[j:j+lc2] -= tmp
i += 1
j -= 1
r = z1[i]
quo[i] = r
tmp = r*z2
z1[i:i+lc2] -= tmp
quo /= scl
rem = z1[i+1:i-1+lc2].copy()
return quo, rem
The provided code snippet includes necessary dependencies for implementing the `_zseries_der` function. Write a Python function `def _zseries_der(zs)` to solve the following problem:
Differentiate a z-series. The derivative is with respect to x, not z. This is achieved using the chain rule and the value of dx/dz given in the module notes. Parameters ---------- zs : z-series The z-series to differentiate. Returns ------- derivative : z-series The derivative Notes ----- The zseries for x (ns) has been multiplied by two in order to avoid using floats that are incompatible with Decimal and likely other specialized scalar types. This scaling has been compensated by multiplying the value of zs by two also so that the two cancels in the division.
Here is the function:
def _zseries_der(zs):
"""Differentiate a z-series.
The derivative is with respect to x, not z. This is achieved using the
chain rule and the value of dx/dz given in the module notes.
Parameters
----------
zs : z-series
The z-series to differentiate.
Returns
-------
derivative : z-series
The derivative
Notes
-----
The zseries for x (ns) has been multiplied by two in order to avoid
using floats that are incompatible with Decimal and likely other
specialized scalar types. This scaling has been compensated by
multiplying the value of zs by two also so that the two cancels in the
division.
"""
n = len(zs)//2
ns = np.array([-1, 0, 1], dtype=zs.dtype)
zs *= np.arange(-n, n+1)*2
d, r = _zseries_div(zs, ns)
return d | Differentiate a z-series. The derivative is with respect to x, not z. This is achieved using the chain rule and the value of dx/dz given in the module notes. Parameters ---------- zs : z-series The z-series to differentiate. Returns ------- derivative : z-series The derivative Notes ----- The zseries for x (ns) has been multiplied by two in order to avoid using floats that are incompatible with Decimal and likely other specialized scalar types. This scaling has been compensated by multiplying the value of zs by two also so that the two cancels in the division. |
169,776 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def _zseries_mul(z1, z2):
"""Multiply two z-series.
Multiply two z-series to produce a z-series.
Parameters
----------
z1, z2 : 1-D ndarray
The arrays must be 1-D but this is not checked.
Returns
-------
product : 1-D ndarray
The product z-series.
Notes
-----
This is simply convolution. If symmetric/anti-symmetric z-series are
denoted by S/A then the following rules apply:
S*S, A*A -> S
S*A, A*S -> A
"""
return np.convolve(z1, z2)
The provided code snippet includes necessary dependencies for implementing the `_zseries_int` function. Write a Python function `def _zseries_int(zs)` to solve the following problem:
Integrate a z-series. The integral is with respect to x, not z. This is achieved by a change of variable using dx/dz given in the module notes. Parameters ---------- zs : z-series The z-series to integrate Returns ------- integral : z-series The indefinite integral Notes ----- The zseries for x (ns) has been multiplied by two in order to avoid using floats that are incompatible with Decimal and likely other specialized scalar types. This scaling has been compensated by dividing the resulting zs by two.
Here is the function:
def _zseries_int(zs):
"""Integrate a z-series.
The integral is with respect to x, not z. This is achieved by a change
of variable using dx/dz given in the module notes.
Parameters
----------
zs : z-series
The z-series to integrate
Returns
-------
integral : z-series
The indefinite integral
Notes
-----
The zseries for x (ns) has been multiplied by two in order to avoid
using floats that are incompatible with Decimal and likely other
specialized scalar types. This scaling has been compensated by
dividing the resulting zs by two.
"""
n = 1 + len(zs)//2
ns = np.array([-1, 0, 1], dtype=zs.dtype)
zs = _zseries_mul(zs, ns)
div = np.arange(-n, n+1)*2
zs[:n] /= div[:n]
zs[n+1:] /= div[n+1:]
zs[n] = 0
return zs | Integrate a z-series. The integral is with respect to x, not z. This is achieved by a change of variable using dx/dz given in the module notes. Parameters ---------- zs : z-series The z-series to integrate Returns ------- integral : z-series The indefinite integral Notes ----- The zseries for x (ns) has been multiplied by two in order to avoid using floats that are incompatible with Decimal and likely other specialized scalar types. This scaling has been compensated by dividing the resulting zs by two. |
169,777 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebadd(c1, c2):
"""
Add one Chebyshev series to another.
Returns the sum of two Chebyshev series `c1` + `c2`. The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the Chebyshev series of their sum.
See Also
--------
chebsub, chebmulx, chebmul, chebdiv, chebpow
Notes
-----
Unlike multiplication, division, etc., the sum of two Chebyshev series
is a Chebyshev series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebadd(c1,c2)
array([4., 4., 4.])
"""
return pu._add(c1, c2)
def chebmulx(c):
"""Multiply a Chebyshev series by x.
Multiply the polynomial `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
Notes
-----
.. versionadded:: 1.5.0
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> C.chebmulx([1,2,3])
array([1. , 2.5, 1. , 1.5])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]*0
prd[1] = c[0]
if len(c) > 1:
tmp = c[1:]/2
prd[2:] = tmp
prd[0:-2] += tmp
return prd
The provided code snippet includes necessary dependencies for implementing the `poly2cheb` function. Write a Python function `def poly2cheb(pol)` to solve the following problem:
Convert a polynomial to a Chebyshev series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Chebyshev series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Chebyshev series. See Also -------- cheb2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy import polynomial as P >>> p = P.Polynomial(range(4)) >>> p Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) >>> c = p.convert(kind=P.Chebyshev) >>> c Chebyshev([1. , 3.25, 1. , 0.75], domain=[-1., 1.], window=[-1., 1.]) >>> P.chebyshev.poly2cheb(range(4)) array([1. , 3.25, 1. , 0.75])
Here is the function:
def poly2cheb(pol):
"""
Convert a polynomial to a Chebyshev series.
Convert an array representing the coefficients of a polynomial (relative
to the "standard" basis) ordered from lowest degree to highest, to an
array of the coefficients of the equivalent Chebyshev series, ordered
from lowest to highest degree.
Parameters
----------
pol : array_like
1-D array containing the polynomial coefficients
Returns
-------
c : ndarray
1-D array containing the coefficients of the equivalent Chebyshev
series.
See Also
--------
cheb2poly
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy import polynomial as P
>>> p = P.Polynomial(range(4))
>>> p
Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
>>> c = p.convert(kind=P.Chebyshev)
>>> c
Chebyshev([1. , 3.25, 1. , 0.75], domain=[-1., 1.], window=[-1., 1.])
>>> P.chebyshev.poly2cheb(range(4))
array([1. , 3.25, 1. , 0.75])
"""
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
for i in range(deg, -1, -1):
res = chebadd(chebmulx(res), pol[i])
return res | Convert a polynomial to a Chebyshev series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Chebyshev series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Chebyshev series. See Also -------- cheb2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy import polynomial as P >>> p = P.Polynomial(range(4)) >>> p Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) >>> c = p.convert(kind=P.Chebyshev) >>> c Chebyshev([1. , 3.25, 1. , 0.75], domain=[-1., 1.], window=[-1., 1.]) >>> P.chebyshev.poly2cheb(range(4)) array([1. , 3.25, 1. , 0.75]) |
169,778 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyadd(c1, c2):
"""
Add one polynomial to another.
Returns the sum of two polynomials `c1` + `c2`. The arguments are
sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to high.
Returns
-------
out : ndarray
The coefficient array representing their sum.
See Also
--------
polysub, polymulx, polymul, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> sum = P.polyadd(c1,c2); sum
array([4., 4., 4.])
>>> P.polyval(2, sum) # 4 + 4(2) + 4(2**2)
28.0
"""
return pu._add(c1, c2)
def polysub(c1, c2):
"""
Subtract one polynomial from another.
Returns the difference of two polynomials `c1` - `c2`. The arguments
are sequences of coefficients from lowest order term to highest, i.e.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of coefficients representing their difference.
See Also
--------
polyadd, polymulx, polymul, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polysub(c1,c2)
array([-2., 0., 2.])
>>> P.polysub(c2,c1) # -P.polysub(c1,c2)
array([ 2., 0., -2.])
"""
return pu._sub(c1, c2)
def polymulx(c):
"""Multiply a polynomial by x.
Multiply the polynomial `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of polynomial coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
See Also
--------
polyadd, polysub, polymul, polydiv, polypow
Notes
-----
.. versionadded:: 1.5.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]*0
prd[1:] = c
return prd
The provided code snippet includes necessary dependencies for implementing the `cheb2poly` function. Write a Python function `def cheb2poly(c)` to solve the following problem:
Convert a Chebyshev series to a polynomial. Convert an array representing the coefficients of a Chebyshev series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- c : array_like 1-D array containing the Chebyshev series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2cheb Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy import polynomial as P >>> c = P.Chebyshev(range(4)) >>> c Chebyshev([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) >>> p = c.convert(kind=P.Polynomial) >>> p Polynomial([-2., -8., 4., 12.], domain=[-1., 1.], window=[-1., 1.]) >>> P.chebyshev.cheb2poly(range(4)) array([-2., -8., 4., 12.])
Here is the function:
def cheb2poly(c):
"""
Convert a Chebyshev series to a polynomial.
Convert an array representing the coefficients of a Chebyshev series,
ordered from lowest degree to highest, to an array of the coefficients
of the equivalent polynomial (relative to the "standard" basis) ordered
from lowest to highest degree.
Parameters
----------
c : array_like
1-D array containing the Chebyshev series coefficients, ordered
from lowest order term to highest.
Returns
-------
pol : ndarray
1-D array containing the coefficients of the equivalent polynomial
(relative to the "standard" basis) ordered from lowest order term
to highest.
See Also
--------
poly2cheb
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy import polynomial as P
>>> c = P.Chebyshev(range(4))
>>> c
Chebyshev([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
>>> p = c.convert(kind=P.Polynomial)
>>> p
Polynomial([-2., -8., 4., 12.], domain=[-1., 1.], window=[-1., 1.])
>>> P.chebyshev.cheb2poly(range(4))
array([-2., -8., 4., 12.])
"""
from .polynomial import polyadd, polysub, polymulx
[c] = pu.as_series([c])
n = len(c)
if n < 3:
return c
else:
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], c1)
c1 = polyadd(tmp, polymulx(c1)*2)
return polyadd(c0, polymulx(c1)) | Convert a Chebyshev series to a polynomial. Convert an array representing the coefficients of a Chebyshev series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest to highest degree. Parameters ---------- c : array_like 1-D array containing the Chebyshev series coefficients, ordered from lowest order term to highest. Returns ------- pol : ndarray 1-D array containing the coefficients of the equivalent polynomial (relative to the "standard" basis) ordered from lowest order term to highest. See Also -------- poly2cheb Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy import polynomial as P >>> c = P.Chebyshev(range(4)) >>> c Chebyshev([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) >>> p = c.convert(kind=P.Polynomial) >>> p Polynomial([-2., -8., 4., 12.], domain=[-1., 1.], window=[-1., 1.]) >>> P.chebyshev.cheb2poly(range(4)) array([-2., -8., 4., 12.]) |
169,779 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebline(off, scl):
"""
Chebyshev series whose graph is a straight line.
Parameters
----------
off, scl : scalars
The specified line is given by ``off + scl*x``.
Returns
-------
y : ndarray
This module's representation of the Chebyshev series for
``off + scl*x``.
See Also
--------
numpy.polynomial.polynomial.polyline
numpy.polynomial.legendre.legline
numpy.polynomial.laguerre.lagline
numpy.polynomial.hermite.hermline
numpy.polynomial.hermite_e.hermeline
Examples
--------
>>> import numpy.polynomial.chebyshev as C
>>> C.chebline(3,2)
array([3, 2])
>>> C.chebval(-3, C.chebline(3,2)) # should be -3
-3.0
"""
if scl != 0:
return np.array([off, scl])
else:
return np.array([off])
def chebmul(c1, c2):
"""
Multiply one Chebyshev series by another.
Returns the product of two Chebyshev series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Chebyshev series coefficients representing their product.
See Also
--------
chebadd, chebsub, chebmulx, chebdiv, chebpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Chebyshev polynomial basis set. Thus, to express
the product as a C-series, it is typically necessary to "reproject"
the product onto said basis set, which typically produces
"unintuitive live" (but correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebmul(c1,c2) # multiplication requires "reprojection"
array([ 6.5, 12. , 12. , 4. , 1.5])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
prd = _zseries_mul(z1, z2)
ret = _zseries_to_cseries(prd)
return pu.trimseq(ret)
The provided code snippet includes necessary dependencies for implementing the `chebfromroots` function. Write a Python function `def chebfromroots(roots)` to solve the following problem:
Generate a Chebyshev series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in Chebyshev form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x) The coefficient of the last term is not generally 1 for monic polynomials in Chebyshev form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below). See Also -------- numpy.polynomial.polynomial.polyfromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.laguerre.lagfromroots numpy.polynomial.hermite.hermfromroots numpy.polynomial.hermite_e.hermefromroots Examples -------- >>> import numpy.polynomial.chebyshev as C >>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis array([ 0. , -0.25, 0. , 0.25]) >>> j = complex(0,1) >>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis array([1.5+0.j, 0. +0.j, 0.5+0.j])
Here is the function:
def chebfromroots(roots):
"""
Generate a Chebyshev series with given roots.
The function returns the coefficients of the polynomial
.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
in Chebyshev form, where the `r_n` are the roots specified in `roots`.
If a zero has multiplicity n, then it must appear in `roots` n times.
For instance, if 2 is a root of multiplicity three and 3 is a root of
multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
roots can appear in any order.
If the returned coefficients are `c`, then
.. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x)
The coefficient of the last term is not generally 1 for monic
polynomials in Chebyshev form.
Parameters
----------
roots : array_like
Sequence containing the roots.
Returns
-------
out : ndarray
1-D array of coefficients. If all roots are real then `out` is a
real array, if some of the roots are complex, then `out` is complex
even if all the coefficients in the result are real (see Examples
below).
See Also
--------
numpy.polynomial.polynomial.polyfromroots
numpy.polynomial.legendre.legfromroots
numpy.polynomial.laguerre.lagfromroots
numpy.polynomial.hermite.hermfromroots
numpy.polynomial.hermite_e.hermefromroots
Examples
--------
>>> import numpy.polynomial.chebyshev as C
>>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis
array([ 0. , -0.25, 0. , 0.25])
>>> j = complex(0,1)
>>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis
array([1.5+0.j, 0. +0.j, 0.5+0.j])
"""
return pu._fromroots(chebline, chebmul, roots) | Generate a Chebyshev series with given roots. The function returns the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), in Chebyshev form, where the `r_n` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x) The coefficient of the last term is not generally 1 for monic polynomials in Chebyshev form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of coefficients. If all roots are real then `out` is a real array, if some of the roots are complex, then `out` is complex even if all the coefficients in the result are real (see Examples below). See Also -------- numpy.polynomial.polynomial.polyfromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.laguerre.lagfromroots numpy.polynomial.hermite.hermfromroots numpy.polynomial.hermite_e.hermefromroots Examples -------- >>> import numpy.polynomial.chebyshev as C >>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis array([ 0. , -0.25, 0. , 0.25]) >>> j = complex(0,1) >>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis array([1.5+0.j, 0. +0.j, 0.5+0.j]) |
169,780 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `chebsub` function. Write a Python function `def chebsub(c1, c2)` to solve the following problem:
Subtract one Chebyshev series from another. Returns the difference of two Chebyshev series `c1` - `c2`. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Chebyshev series coefficients ordered from low to high. Returns ------- out : ndarray Of Chebyshev series coefficients representing their difference. See Also -------- chebadd, chebmulx, chebmul, chebdiv, chebpow Notes ----- Unlike multiplication, division, etc., the difference of two Chebyshev series is a Chebyshev series (without having to "reproject" the result onto the basis set) so subtraction, just like that of "standard" polynomials, is simply "component-wise." Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> C.chebsub(c1,c2) array([-2., 0., 2.]) >>> C.chebsub(c2,c1) # -C.chebsub(c1,c2) array([ 2., 0., -2.])
Here is the function:
def chebsub(c1, c2):
"""
Subtract one Chebyshev series from another.
Returns the difference of two Chebyshev series `c1` - `c2`. The
sequences of coefficients are from lowest order term to highest, i.e.,
[1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Chebyshev series coefficients representing their difference.
See Also
--------
chebadd, chebmulx, chebmul, chebdiv, chebpow
Notes
-----
Unlike multiplication, division, etc., the difference of two Chebyshev
series is a Chebyshev series (without having to "reproject" the result
onto the basis set) so subtraction, just like that of "standard"
polynomials, is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebsub(c1,c2)
array([-2., 0., 2.])
>>> C.chebsub(c2,c1) # -C.chebsub(c1,c2)
array([ 2., 0., -2.])
"""
return pu._sub(c1, c2) | Subtract one Chebyshev series from another. Returns the difference of two Chebyshev series `c1` - `c2`. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Chebyshev series coefficients ordered from low to high. Returns ------- out : ndarray Of Chebyshev series coefficients representing their difference. See Also -------- chebadd, chebmulx, chebmul, chebdiv, chebpow Notes ----- Unlike multiplication, division, etc., the difference of two Chebyshev series is a Chebyshev series (without having to "reproject" the result onto the basis set) so subtraction, just like that of "standard" polynomials, is simply "component-wise." Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> C.chebsub(c1,c2) array([-2., 0., 2.]) >>> C.chebsub(c2,c1) # -C.chebsub(c1,c2) array([ 2., 0., -2.]) |
169,781 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def _cseries_to_zseries(c):
"""Convert Chebyshev series to z-series.
Convert a Chebyshev series to the equivalent z-series. The result is
never an empty array. The dtype of the return is the same as that of
the input. No checks are run on the arguments as this routine is for
internal use.
Parameters
----------
c : 1-D ndarray
Chebyshev coefficients, ordered from low to high
Returns
-------
zs : 1-D ndarray
Odd length symmetric z-series, ordered from low to high.
"""
n = c.size
zs = np.zeros(2*n-1, dtype=c.dtype)
zs[n-1:] = c/2
return zs + zs[::-1]
def _zseries_to_cseries(zs):
"""Convert z-series to a Chebyshev series.
Convert a z series to the equivalent Chebyshev series. The result is
never an empty array. The dtype of the return is the same as that of
the input. No checks are run on the arguments as this routine is for
internal use.
Parameters
----------
zs : 1-D ndarray
Odd length symmetric z-series, ordered from low to high.
Returns
-------
c : 1-D ndarray
Chebyshev coefficients, ordered from low to high.
"""
n = (zs.size + 1)//2
c = zs[n-1:].copy()
c[1:n] *= 2
return c
def _zseries_div(z1, z2):
"""Divide the first z-series by the second.
Divide `z1` by `z2` and return the quotient and remainder as z-series.
Warning: this implementation only applies when both z1 and z2 have the
same symmetry, which is sufficient for present purposes.
Parameters
----------
z1, z2 : 1-D ndarray
The arrays must be 1-D and have the same symmetry, but this is not
checked.
Returns
-------
(quotient, remainder) : 1-D ndarrays
Quotient and remainder as z-series.
Notes
-----
This is not the same as polynomial division on account of the desired form
of the remainder. If symmetric/anti-symmetric z-series are denoted by S/A
then the following rules apply:
S/S -> S,S
A/A -> S,A
The restriction to types of the same symmetry could be fixed but seems like
unneeded generality. There is no natural form for the remainder in the case
where there is no symmetry.
"""
z1 = z1.copy()
z2 = z2.copy()
lc1 = len(z1)
lc2 = len(z2)
if lc2 == 1:
z1 /= z2
return z1, z1[:1]*0
elif lc1 < lc2:
return z1[:1]*0, z1
else:
dlen = lc1 - lc2
scl = z2[0]
z2 /= scl
quo = np.empty(dlen + 1, dtype=z1.dtype)
i = 0
j = dlen
while i < j:
r = z1[i]
quo[i] = z1[i]
quo[dlen - i] = r
tmp = r*z2
z1[i:i+lc2] -= tmp
z1[j:j+lc2] -= tmp
i += 1
j -= 1
r = z1[i]
quo[i] = r
tmp = r*z2
z1[i:i+lc2] -= tmp
quo /= scl
rem = z1[i+1:i-1+lc2].copy()
return quo, rem
The provided code snippet includes necessary dependencies for implementing the `chebdiv` function. Write a Python function `def chebdiv(c1, c2)` to solve the following problem:
Divide one Chebyshev series by another. Returns the quotient-with-remainder of two Chebyshev series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Chebyshev series coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of Chebyshev series coefficients representing the quotient and remainder. See Also -------- chebadd, chebsub, chebmulx, chebmul, chebpow Notes ----- In general, the (polynomial) division of one C-series by another results in quotient and remainder terms that are not in the Chebyshev polynomial basis set. Thus, to express these results as C-series, it is typically necessary to "reproject" the results onto said basis set, which typically produces "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not (array([3.]), array([-8., -4.])) >>> c2 = (0,1,2,3) >>> C.chebdiv(c2,c1) # neither "intuitive" (array([0., 2.]), array([-2., -4.]))
Here is the function:
def chebdiv(c1, c2):
"""
Divide one Chebyshev series by another.
Returns the quotient-with-remainder of two Chebyshev series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarrays
Of Chebyshev series coefficients representing the quotient and
remainder.
See Also
--------
chebadd, chebsub, chebmulx, chebmul, chebpow
Notes
-----
In general, the (polynomial) division of one C-series by another
results in quotient and remainder terms that are not in the Chebyshev
polynomial basis set. Thus, to express these results as C-series, it
is typically necessary to "reproject" the results onto said basis
set, which typically produces "unintuitive" (but correct) results;
see Examples section below.
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not
(array([3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> C.chebdiv(c2,c1) # neither "intuitive"
(array([0., 2.]), array([-2., -4.]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0:
raise ZeroDivisionError()
# note: this is more efficient than `pu._div(chebmul, c1, c2)`
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2:
return c1[:1]*0, c1
elif lc2 == 1:
return c1/c2[-1], c1[:1]*0
else:
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
quo, rem = _zseries_div(z1, z2)
quo = pu.trimseq(_zseries_to_cseries(quo))
rem = pu.trimseq(_zseries_to_cseries(rem))
return quo, rem | Divide one Chebyshev series by another. Returns the quotient-with-remainder of two Chebyshev series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Chebyshev series coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of Chebyshev series coefficients representing the quotient and remainder. See Also -------- chebadd, chebsub, chebmulx, chebmul, chebpow Notes ----- In general, the (polynomial) division of one C-series by another results in quotient and remainder terms that are not in the Chebyshev polynomial basis set. Thus, to express these results as C-series, it is typically necessary to "reproject" the results onto said basis set, which typically produces "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not (array([3.]), array([-8., -4.])) >>> c2 = (0,1,2,3) >>> C.chebdiv(c2,c1) # neither "intuitive" (array([0., 2.]), array([-2., -4.])) |
169,782 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def _cseries_to_zseries(c):
"""Convert Chebyshev series to z-series.
Convert a Chebyshev series to the equivalent z-series. The result is
never an empty array. The dtype of the return is the same as that of
the input. No checks are run on the arguments as this routine is for
internal use.
Parameters
----------
c : 1-D ndarray
Chebyshev coefficients, ordered from low to high
Returns
-------
zs : 1-D ndarray
Odd length symmetric z-series, ordered from low to high.
"""
n = c.size
zs = np.zeros(2*n-1, dtype=c.dtype)
zs[n-1:] = c/2
return zs + zs[::-1]
def _zseries_to_cseries(zs):
"""Convert z-series to a Chebyshev series.
Convert a z series to the equivalent Chebyshev series. The result is
never an empty array. The dtype of the return is the same as that of
the input. No checks are run on the arguments as this routine is for
internal use.
Parameters
----------
zs : 1-D ndarray
Odd length symmetric z-series, ordered from low to high.
Returns
-------
c : 1-D ndarray
Chebyshev coefficients, ordered from low to high.
"""
n = (zs.size + 1)//2
c = zs[n-1:].copy()
c[1:n] *= 2
return c
The provided code snippet includes necessary dependencies for implementing the `chebpow` function. Write a Python function `def chebpow(c, pow, maxpower=16)` to solve the following problem:
Raise a Chebyshev series to a power. Returns the Chebyshev series `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``T_0 + 2*T_1 + 3*T_2.`` Parameters ---------- c : array_like 1-D array of Chebyshev series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Chebyshev series of power. See Also -------- chebadd, chebsub, chebmulx, chebmul, chebdiv Examples -------- >>> from numpy.polynomial import chebyshev as C >>> C.chebpow([1, 2, 3, 4], 2) array([15.5, 22. , 16. , ..., 12.5, 12. , 8. ])
Here is the function:
def chebpow(c, pow, maxpower=16):
"""Raise a Chebyshev series to a power.
Returns the Chebyshev series `c` raised to the power `pow`. The
argument `c` is a sequence of coefficients ordered from low to high.
i.e., [1,2,3] is the series ``T_0 + 2*T_1 + 3*T_2.``
Parameters
----------
c : array_like
1-D array of Chebyshev series coefficients ordered from low to
high.
pow : integer
Power to which the series will be raised
maxpower : integer, optional
Maximum power allowed. This is mainly to limit growth of the series
to unmanageable size. Default is 16
Returns
-------
coef : ndarray
Chebyshev series of power.
See Also
--------
chebadd, chebsub, chebmulx, chebmul, chebdiv
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> C.chebpow([1, 2, 3, 4], 2)
array([15.5, 22. , 16. , ..., 12.5, 12. , 8. ])
"""
# note: this is more efficient than `pu._pow(chebmul, c1, c2)`, as it
# avoids converting between z and c series repeatedly
# c is a trimmed copy
[c] = pu.as_series([c])
power = int(pow)
if power != pow or power < 0:
raise ValueError("Power must be a non-negative integer.")
elif maxpower is not None and power > maxpower:
raise ValueError("Power is too large")
elif power == 0:
return np.array([1], dtype=c.dtype)
elif power == 1:
return c
else:
# This can be made more efficient by using powers of two
# in the usual way.
zs = _cseries_to_zseries(c)
prd = zs
for i in range(2, power + 1):
prd = np.convolve(prd, zs)
return _zseries_to_cseries(prd) | Raise a Chebyshev series to a power. Returns the Chebyshev series `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``T_0 + 2*T_1 + 3*T_2.`` Parameters ---------- c : array_like 1-D array of Chebyshev series coefficients ordered from low to high. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Chebyshev series of power. See Also -------- chebadd, chebsub, chebmulx, chebmul, chebdiv Examples -------- >>> from numpy.polynomial import chebyshev as C >>> C.chebpow([1, 2, 3, 4], 2) array([15.5, 22. , 16. , ..., 12.5, 12. , 8. ]) |
169,783 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `chebder` function. Write a Python function `def chebder(c, m=1, scl=1, axis=0)` to solve the following problem:
Differentiate a Chebyshev series. Returns the Chebyshev series coefficients `c` differentiated `m` times along `axis`. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``1*T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by `scl`. The end result is multiplication by ``scl**m``. This is for use in a linear change of variable. (Default: 1) axis : int, optional Axis over which the derivative is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- der : ndarray Chebyshev series of the derivative. See Also -------- chebint Notes ----- In general, the result of differentiating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c = (1,2,3,4) >>> C.chebder(c) array([14., 12., 24.]) >>> C.chebder(c,3) array([96.]) >>> C.chebder(c,scl=-1) array([-14., -12., -24.]) >>> C.chebder(c,2,-1) array([12., 96.])
Here is the function:
def chebder(c, m=1, scl=1, axis=0):
"""
Differentiate a Chebyshev series.
Returns the Chebyshev series coefficients `c` differentiated `m` times
along `axis`. At each iteration the result is multiplied by `scl` (the
scaling factor is for use in a linear change of variable). The argument
`c` is an array of coefficients from low to high degree along each
axis, e.g., [1,2,3] represents the series ``1*T_0 + 2*T_1 + 3*T_2``
while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) +
2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is
``y``.
Parameters
----------
c : array_like
Array of Chebyshev series coefficients. If c is multidimensional
the different axis correspond to different variables with the
degree in each axis given by the corresponding index.
m : int, optional
Number of derivatives taken, must be non-negative. (Default: 1)
scl : scalar, optional
Each differentiation is multiplied by `scl`. The end result is
multiplication by ``scl**m``. This is for use in a linear change of
variable. (Default: 1)
axis : int, optional
Axis over which the derivative is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
der : ndarray
Chebyshev series of the derivative.
See Also
--------
chebint
Notes
-----
In general, the result of differentiating a C-series needs to be
"reprojected" onto the C-series basis set. Thus, typically, the
result of this function is "unintuitive," albeit correct; see Examples
section below.
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c = (1,2,3,4)
>>> C.chebder(c)
array([14., 12., 24.])
>>> C.chebder(c,3)
array([96.])
>>> C.chebder(c,scl=-1)
array([-14., -12., -24.])
>>> C.chebder(c,2,-1)
array([12., 96.])
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
cnt = pu._deprecate_as_int(m, "the order of derivation")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of derivation must be non-negative")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
n = len(c)
if cnt >= n:
c = c[:1]*0
else:
for i in range(cnt):
n = n - 1
c *= scl
der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
for j in range(n, 2, -1):
der[j - 1] = (2*j)*c[j]
c[j - 2] += (j*c[j])/(j - 2)
if n > 1:
der[1] = 4*c[2]
der[0] = c[1]
c = der
c = np.moveaxis(c, 0, iaxis)
return c | Differentiate a Chebyshev series. Returns the Chebyshev series coefficients `c` differentiated `m` times along `axis`. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``1*T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by `scl`. The end result is multiplication by ``scl**m``. This is for use in a linear change of variable. (Default: 1) axis : int, optional Axis over which the derivative is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- der : ndarray Chebyshev series of the derivative. See Also -------- chebint Notes ----- In general, the result of differentiating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c = (1,2,3,4) >>> C.chebder(c) array([14., 12., 24.]) >>> C.chebder(c,3) array([96.]) >>> C.chebder(c,scl=-1) array([-14., -12., -24.]) >>> C.chebder(c,2,-1) array([12., 96.]) |
169,784 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebval(x, c, tensor=True):
"""
Evaluate a Chebyshev series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
chebval2d, chebgrid2d, chebval3d, chebgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
x2 = 2*x
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
c0 = c[-i] - c1
c1 = tmp + c1*x2
return c0 + c1*x
The provided code snippet includes necessary dependencies for implementing the `chebint` function. Write a Python function `def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0)` to solve the following problem:
Integrate a Chebyshev series. Returns the Chebyshev series coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at zero is the first value in the list, the value of the second integral at zero is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray C-series coefficients of the integral. Raises ------ ValueError If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- chebder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a`- perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c = (1,2,3) >>> C.chebint(c) array([ 0.5, -0.5, 0.5, 0.5]) >>> C.chebint(c,3) array([ 0.03125 , -0.1875 , 0.04166667, -0.05208333, 0.01041667, # may vary 0.00625 ]) >>> C.chebint(c, k=3) array([ 3.5, -0.5, 0.5, 0.5]) >>> C.chebint(c,lbnd=-2) array([ 8.5, -0.5, 0.5, 0.5]) >>> C.chebint(c,scl=-2) array([-1., 1., -1., -1.])
Here is the function:
def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
"""
Integrate a Chebyshev series.
Returns the Chebyshev series coefficients `c` integrated `m` times from
`lbnd` along `axis`. At each iteration the resulting series is
**multiplied** by `scl` and an integration constant, `k`, is added.
The scaling factor is for use in a linear change of variable. ("Buyer
beware": note that, depending on what one is doing, one may want `scl`
to be the reciprocal of what one might expect; for more information,
see the Notes section below.) The argument `c` is an array of
coefficients from low to high degree along each axis, e.g., [1,2,3]
represents the series ``T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]]
represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) +
2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
Parameters
----------
c : array_like
Array of Chebyshev series coefficients. If c is multidimensional
the different axis correspond to different variables with the
degree in each axis given by the corresponding index.
m : int, optional
Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
Integration constant(s). The value of the first integral at zero
is the first value in the list, the value of the second integral
at zero is the second value, etc. If ``k == []`` (the default),
all constants are set to zero. If ``m == 1``, a single scalar can
be given instead of a list.
lbnd : scalar, optional
The lower bound of the integral. (Default: 0)
scl : scalar, optional
Following each integration the result is *multiplied* by `scl`
before the integration constant is added. (Default: 1)
axis : int, optional
Axis over which the integral is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
S : ndarray
C-series coefficients of the integral.
Raises
------
ValueError
If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
``np.ndim(scl) != 0``.
See Also
--------
chebder
Notes
-----
Note that the result of each integration is *multiplied* by `scl`.
Why is this important to note? Say one is making a linear change of
variable :math:`u = ax + b` in an integral relative to `x`. Then
:math:`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a`- perhaps not what one would have first thought.
Also note that, in general, the result of integrating a C-series needs
to be "reprojected" onto the C-series basis set. Thus, typically,
the result of this function is "unintuitive," albeit correct; see
Examples section below.
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c = (1,2,3)
>>> C.chebint(c)
array([ 0.5, -0.5, 0.5, 0.5])
>>> C.chebint(c,3)
array([ 0.03125 , -0.1875 , 0.04166667, -0.05208333, 0.01041667, # may vary
0.00625 ])
>>> C.chebint(c, k=3)
array([ 3.5, -0.5, 0.5, 0.5])
>>> C.chebint(c,lbnd=-2)
array([ 8.5, -0.5, 0.5, 0.5])
>>> C.chebint(c,scl=-2)
array([-1., 1., -1., -1.])
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if not np.iterable(k):
k = [k]
cnt = pu._deprecate_as_int(m, "the order of integration")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of integration must be non-negative")
if len(k) > cnt:
raise ValueError("Too many integration constants")
if np.ndim(lbnd) != 0:
raise ValueError("lbnd must be a scalar.")
if np.ndim(scl) != 0:
raise ValueError("scl must be a scalar.")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
k = list(k) + [0]*(cnt - len(k))
for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
c[0] += k[i]
else:
tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
tmp[0] = c[0]*0
tmp[1] = c[0]
if n > 1:
tmp[2] = c[1]/4
for j in range(2, n):
tmp[j + 1] = c[j]/(2*(j + 1))
tmp[j - 1] -= c[j]/(2*(j - 1))
tmp[0] += k[i] - chebval(lbnd, tmp)
c = tmp
c = np.moveaxis(c, 0, iaxis)
return c | Integrate a Chebyshev series. Returns the Chebyshev series coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at zero is the first value in the list, the value of the second integral at zero is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray C-series coefficients of the integral. Raises ------ ValueError If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- chebder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a`- perhaps not what one would have first thought. Also note that, in general, the result of integrating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c = (1,2,3) >>> C.chebint(c) array([ 0.5, -0.5, 0.5, 0.5]) >>> C.chebint(c,3) array([ 0.03125 , -0.1875 , 0.04166667, -0.05208333, 0.01041667, # may vary 0.00625 ]) >>> C.chebint(c, k=3) array([ 3.5, -0.5, 0.5, 0.5]) >>> C.chebint(c,lbnd=-2) array([ 8.5, -0.5, 0.5, 0.5]) >>> C.chebint(c,scl=-2) array([-1., 1., -1., -1.]) |
169,785 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebval(x, c, tensor=True):
"""
Evaluate a Chebyshev series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
chebval2d, chebgrid2d, chebval3d, chebgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
x2 = 2*x
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
c0 = c[-i] - c1
c1 = tmp + c1*x2
return c0 + c1*x
The provided code snippet includes necessary dependencies for implementing the `chebval2d` function. Write a Python function `def chebval2d(x, y, c)` to solve the following problem:
Evaluate a 2-D Chebyshev series at points (x, y). This function returns the values: .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * T_i(x) * T_j(y) The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension greater than 2 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Chebyshev series at points formed from pairs of corresponding values from `x` and `y`. See Also -------- chebval, chebgrid2d, chebval3d, chebgrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def chebval2d(x, y, c):
"""
Evaluate a 2-D Chebyshev series at points (x, y).
This function returns the values:
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * T_i(x) * T_j(y)
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars and they
must have the same shape after conversion. In either case, either `x`
and `y` or their elements must support multiplication and addition both
with themselves and with the elements of `c`.
If `c` is a 1-D array a one is implicitly appended to its shape to make
it 2-D. The shape of the result will be c.shape[2:] + x.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points `(x, y)`,
where `x` and `y` must have the same shape. If `x` or `y` is a list
or tuple, it is first converted to an ndarray, otherwise it is left
unchanged and if it isn't an ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term
of multi-degree i,j is contained in ``c[i,j]``. If `c` has
dimension greater than 2 the remaining indices enumerate multiple
sets of coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional Chebyshev series at points formed
from pairs of corresponding values from `x` and `y`.
See Also
--------
chebval, chebgrid2d, chebval3d, chebgrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(chebval, c, x, y) | Evaluate a 2-D Chebyshev series at points (x, y). This function returns the values: .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * T_i(x) * T_j(y) The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in ``c[i,j]``. If `c` has dimension greater than 2 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Chebyshev series at points formed from pairs of corresponding values from `x` and `y`. See Also -------- chebval, chebgrid2d, chebval3d, chebgrid3d Notes ----- .. versionadded:: 1.7.0 |
169,786 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebval(x, c, tensor=True):
"""
Evaluate a Chebyshev series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
chebval2d, chebgrid2d, chebval3d, chebgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
x2 = 2*x
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
c0 = c[-i] - c1
c1 = tmp + c1*x2
return c0 + c1*x
The provided code snippet includes necessary dependencies for implementing the `chebgrid2d` function. Write a Python function `def chebgrid2d(x, y, c)` to solve the following problem:
Evaluate a 2-D Chebyshev series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * T_i(a) * T_j(b), where the points `(a, b)` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Chebyshev series at points in the Cartesian product of `x` and `y`. See Also -------- chebval, chebval2d, chebval3d, chebgrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def chebgrid2d(x, y, c):
"""
Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.
This function returns the values:
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * T_i(a) * T_j(b),
where the points `(a, b)` consist of all pairs formed by taking
`a` from `x` and `b` from `y`. The resulting points form a grid with
`x` in the first dimension and `y` in the second.
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars. In either
case, either `x` and `y` or their elements must support multiplication
and addition both with themselves and with the elements of `c`.
If `c` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape + y.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points in the
Cartesian product of `x` and `y`. If `x` or `y` is a list or
tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional Chebyshev series at points in the
Cartesian product of `x` and `y`.
See Also
--------
chebval, chebval2d, chebval3d, chebgrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(chebval, c, x, y) | Evaluate a 2-D Chebyshev series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * T_i(a) * T_j(b), where the points `(a, b)` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Chebyshev series at points in the Cartesian product of `x` and `y`. See Also -------- chebval, chebval2d, chebval3d, chebgrid3d Notes ----- .. versionadded:: 1.7.0 |
169,787 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebval(x, c, tensor=True):
"""
Evaluate a Chebyshev series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
chebval2d, chebgrid2d, chebval3d, chebgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
x2 = 2*x
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
c0 = c[-i] - c1
c1 = tmp + c1*x2
return c0 + c1*x
The provided code snippet includes necessary dependencies for implementing the `chebval3d` function. Write a Python function `def chebval3d(x, y, z, c)` to solve the following problem:
Evaluate a 3-D Chebyshev series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_k(z) The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- chebval, chebval2d, chebgrid2d, chebgrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def chebval3d(x, y, z, c):
"""
Evaluate a 3-D Chebyshev series at points (x, y, z).
This function returns the values:
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_k(z)
The parameters `x`, `y`, and `z` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
`x`, `y`, and `z` or their elements must support multiplication and
addition both with themselves and with the elements of `c`.
If `c` has fewer than 3 dimensions, ones are implicitly appended to its
shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape.
Parameters
----------
x, y, z : array_like, compatible object
The three dimensional series is evaluated at the points
`(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If
any of `x`, `y`, or `z` is a list or tuple, it is first converted
to an ndarray, otherwise it is left unchanged and if it isn't an
ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
greater than 3 the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the multidimensional polynomial on points formed with
triples of corresponding values from `x`, `y`, and `z`.
See Also
--------
chebval, chebval2d, chebgrid2d, chebgrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(chebval, c, x, y, z) | Evaluate a 3-D Chebyshev series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_k(z) The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- chebval, chebval2d, chebgrid2d, chebgrid3d Notes ----- .. versionadded:: 1.7.0 |
169,788 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebval(x, c, tensor=True):
"""
Evaluate a Chebyshev series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, algebra_like
The shape of the return value is described above.
See Also
--------
chebval2d, chebgrid2d, chebval3d, chebgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
c = c.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
if len(c) == 1:
c0 = c[0]
c1 = 0
elif len(c) == 2:
c0 = c[0]
c1 = c[1]
else:
x2 = 2*x
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
c0 = c[-i] - c1
c1 = tmp + c1*x2
return c0 + c1*x
The provided code snippet includes necessary dependencies for implementing the `chebgrid3d` function. Write a Python function `def chebgrid3d(x, y, z, c)` to solve the following problem:
Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * T_i(a) * T_j(b) * T_k(c) where the points `(a, b, c)` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- chebval, chebval2d, chebgrid2d, chebval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def chebgrid3d(x, y, z, c):
"""
Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z.
This function returns the values:
.. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * T_i(a) * T_j(b) * T_k(c)
where the points `(a, b, c)` consist of all triples formed by taking
`a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
a grid with `x` in the first dimension, `y` in the second, and `z` in
the third.
The parameters `x`, `y`, and `z` are converted to arrays only if they
are tuples or a lists, otherwise they are treated as a scalars. In
either case, either `x`, `y`, and `z` or their elements must support
multiplication and addition both with themselves and with the elements
of `c`.
If `c` has fewer than three dimensions, ones are implicitly appended to
its shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape + y.shape + z.shape.
Parameters
----------
x, y, z : array_like, compatible objects
The three dimensional series is evaluated at the points in the
Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a
list or tuple, it is first converted to an ndarray, otherwise it is
left unchanged and, if it isn't an ndarray, it is treated as a
scalar.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree i,j are contained in ``c[i,j]``. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points in the Cartesian
product of `x` and `y`.
See Also
--------
chebval, chebval2d, chebgrid2d, chebval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(chebval, c, x, y, z) | Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * T_i(a) * T_j(b) * T_k(c) where the points `(a, b, c)` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- chebval, chebval2d, chebgrid2d, chebval3d Notes ----- .. versionadded:: 1.7.0 |
169,789 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = T_i(x),
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Chebyshev polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and
``chebval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Chebyshev series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Chebyshev polynomial. The dtype will be the same as
the converted `x`.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
# Use forward recursion to generate the entries.
v[0] = x*0 + 1
if ideg > 0:
x2 = 2*x
v[1] = x
for i in range(2, ideg + 1):
v[i] = v[i-1]*x2 - v[i-2]
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `chebvander2d` function. Write a Python function `def chebvander2d(x, y, deg)` to solve the following problem:
Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y)`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = T_i(x) * T_j(y), where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of `V` index the points `(x, y)` and the last index encodes the degrees of the Chebyshev polynomials. If ``V = chebvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``chebval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Chebyshev series of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- chebvander, chebvander3d, chebval2d, chebval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def chebvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y)`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (deg[1] + 1)*i + j] = T_i(x) * T_j(y),
where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
`V` index the points `(x, y)` and the last index encodes the degrees of
the Chebyshev polynomials.
If ``V = chebvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
correspond to the elements of a 2-D coefficient array `c` of shape
(xdeg + 1, ydeg + 1) in the order
.. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
and ``np.dot(V, c.flat)`` and ``chebval2d(x, y, c)`` will be the same
up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 2-D Chebyshev
series of the same degrees and sample points.
Parameters
----------
x, y : array_like
Arrays of point coordinates, all of the same shape. The dtypes
will be converted to either float64 or complex128 depending on
whether any of the elements are complex. Scalars are converted to
1-D arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg].
Returns
-------
vander2d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same
as the converted `x` and `y`.
See Also
--------
chebvander, chebvander3d, chebval2d, chebval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((chebvander, chebvander), (x, y), deg) | Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y)`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = T_i(x) * T_j(y), where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of `V` index the points `(x, y)` and the last index encodes the degrees of the Chebyshev polynomials. If ``V = chebvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``chebval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Chebyshev series of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- chebvander, chebvander3d, chebval2d, chebval3d Notes ----- .. versionadded:: 1.7.0 |
169,790 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = T_i(x),
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Chebyshev polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and
``chebval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Chebyshev series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Chebyshev polynomial. The dtype will be the same as
the converted `x`.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
# Use forward recursion to generate the entries.
v[0] = x*0 + 1
if ideg > 0:
x2 = 2*x
v[1] = x
for i in range(2, ideg + 1):
v[i] = v[i-1]*x2 - v[i-2]
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `chebvander3d` function. Write a Python function `def chebvander3d(x, y, z, deg)` to solve the following problem:
Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x)*T_j(y)*T_k(z), where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading indices of `V` index the points `(x, y, z)` and the last index encodes the degrees of the Chebyshev polynomials. If ``V = chebvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``chebval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Chebyshev series of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- chebvander, chebvander3d, chebval2d, chebval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def chebvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
then The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x)*T_j(y)*T_k(z),
where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading
indices of `V` index the points `(x, y, z)` and the last index encodes
the degrees of the Chebyshev polynomials.
If ``V = chebvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
of `V` correspond to the elements of a 3-D coefficient array `c` of
shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
.. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
and ``np.dot(V, c.flat)`` and ``chebval3d(x, y, z, c)`` will be the
same up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 3-D Chebyshev
series of the same degrees and sample points.
Parameters
----------
x, y, z : array_like
Arrays of point coordinates, all of the same shape. The dtypes will
be converted to either float64 or complex128 depending on whether
any of the elements are complex. Scalars are converted to 1-D
arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg, z_deg].
Returns
-------
vander3d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will
be the same as the converted `x`, `y`, and `z`.
See Also
--------
chebvander, chebvander3d, chebval2d, chebval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((chebvander, chebvander, chebvander), (x, y, z), deg) | Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x)*T_j(y)*T_k(z), where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading indices of `V` index the points `(x, y, z)` and the last index encodes the degrees of the Chebyshev polynomials. If ``V = chebvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``chebval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Chebyshev series of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- chebvander, chebvander3d, chebval2d, chebval3d Notes ----- .. versionadded:: 1.7.0 |
169,791 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = T_i(x),
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Chebyshev polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and
``chebval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Chebyshev series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Chebyshev polynomial. The dtype will be the same as
the converted `x`.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
# Use forward recursion to generate the entries.
v[0] = x*0 + 1
if ideg > 0:
x2 = 2*x
v[1] = x
for i in range(2, ideg + 1):
v[i] = v[i-1]*x2 - v[i-2]
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `chebfit` function. Write a Python function `def chebfit(x, y, deg, rcond=None, full=False, w=None)` to solve the following problem:
Least squares fit of Chebyshev series to data. Return the coefficients of a Chebyshev series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x), where `n` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer, all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the weight ``w[i]`` applies to the unsquared residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. When using inverse-variance weighting, use ``w[i] = 1/sigma(y[i])``. The default value is None. .. versionadded:: 1.5.0 Returns ------- coef : ndarray, shape (M,) or (M, K) Chebyshev coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`. [residuals, rank, singular_values, rcond] : list These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the numerical rank of the scaled Vandermonde matrix - singular_values -- singular values of the scaled Vandermonde matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if ``full == False``. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- numpy.polynomial.polynomial.polyfit numpy.polynomial.legendre.legfit numpy.polynomial.laguerre.lagfit numpy.polynomial.hermite.hermfit numpy.polynomial.hermite_e.hermefit chebval : Evaluates a Chebyshev series. chebvander : Vandermonde matrix of Chebyshev series. chebweight : Chebyshev weight function. numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Chebyshev series `p` that minimizes the sum of the weighted squared errors .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, where :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the coefficients to be solved for, `w` are the weights, and `y` are the observed values. This equation is then solved using the singular value decomposition of `V`. If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Chebyshev series are usually better conditioned than fits using power series, but much can depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples --------
Here is the function:
def chebfit(x, y, deg, rcond=None, full=False, w=None):
"""
Least squares fit of Chebyshev series to data.
Return the coefficients of a Chebyshev series of degree `deg` that is the
least squares fit to the data values `y` given at points `x`. If `y` is
1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
fits are done, one for each column of `y`, and the resulting
coefficients are stored in the corresponding columns of a 2-D return.
The fitted polynomial(s) are in the form
.. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x),
where `n` is `deg`.
Parameters
----------
x : array_like, shape (M,)
x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,) or (M, K)
y-coordinates of the sample points. Several data sets of sample
points sharing the same x-coordinates can be fitted at once by
passing in a 2D-array that contains one dataset per column.
deg : int or 1-D array_like
Degree(s) of the fitting polynomials. If `deg` is a single integer,
all terms up to and including the `deg`'th term are included in the
fit. For NumPy versions >= 1.11.0 a list of integers specifying the
degrees of the terms to include may be used instead.
rcond : float, optional
Relative condition number of the fit. Singular values smaller than
this relative to the largest singular value will be ignored. The
default value is len(x)*eps, where eps is the relative precision of
the float type, about 2e-16 in most cases.
full : bool, optional
Switch determining nature of return value. When it is False (the
default) just the coefficients are returned, when True diagnostic
information from the singular value decomposition is also returned.
w : array_like, shape (`M`,), optional
Weights. If not None, the weight ``w[i]`` applies to the unsquared
residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
chosen so that the errors of the products ``w[i]*y[i]`` all have the
same variance. When using inverse-variance weighting, use
``w[i] = 1/sigma(y[i])``. The default value is None.
.. versionadded:: 1.5.0
Returns
-------
coef : ndarray, shape (M,) or (M, K)
Chebyshev coefficients ordered from low to high. If `y` was 2-D,
the coefficients for the data in column k of `y` are in column
`k`.
[residuals, rank, singular_values, rcond] : list
These values are only returned if ``full == True``
- residuals -- sum of squared residuals of the least squares fit
- rank -- the numerical rank of the scaled Vandermonde matrix
- singular_values -- singular values of the scaled Vandermonde matrix
- rcond -- value of `rcond`.
For more details, see `numpy.linalg.lstsq`.
Warns
-----
RankWarning
The rank of the coefficient matrix in the least-squares fit is
deficient. The warning is only raised if ``full == False``. The
warnings can be turned off by
>>> import warnings
>>> warnings.simplefilter('ignore', np.RankWarning)
See Also
--------
numpy.polynomial.polynomial.polyfit
numpy.polynomial.legendre.legfit
numpy.polynomial.laguerre.lagfit
numpy.polynomial.hermite.hermfit
numpy.polynomial.hermite_e.hermefit
chebval : Evaluates a Chebyshev series.
chebvander : Vandermonde matrix of Chebyshev series.
chebweight : Chebyshev weight function.
numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes
-----
The solution is the coefficients of the Chebyshev series `p` that
minimizes the sum of the weighted squared errors
.. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
where :math:`w_j` are the weights. This problem is solved by setting up
as the (typically) overdetermined matrix equation
.. math:: V(x) * c = w * y,
where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
coefficients to be solved for, `w` are the weights, and `y` are the
observed values. This equation is then solved using the singular value
decomposition of `V`.
If some of the singular values of `V` are so small that they are
neglected, then a `RankWarning` will be issued. This means that the
coefficient values may be poorly determined. Using a lower order fit
will usually get rid of the warning. The `rcond` parameter can also be
set to a value smaller than its default, but the resulting fit may be
spurious and have large contributions from roundoff error.
Fits using Chebyshev series are usually better conditioned than fits
using power series, but much can depend on the distribution of the
sample points and the smoothness of the data. If the quality of the fit
is inadequate splines may be a good alternative.
References
----------
.. [1] Wikipedia, "Curve fitting",
https://en.wikipedia.org/wiki/Curve_fitting
Examples
--------
"""
return pu._fit(chebvander, x, y, deg, rcond, full, w) | Least squares fit of Chebyshev series to data. Return the coefficients of a Chebyshev series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x), where `n` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer, all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the weight ``w[i]`` applies to the unsquared residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are chosen so that the errors of the products ``w[i]*y[i]`` all have the same variance. When using inverse-variance weighting, use ``w[i] = 1/sigma(y[i])``. The default value is None. .. versionadded:: 1.5.0 Returns ------- coef : ndarray, shape (M,) or (M, K) Chebyshev coefficients ordered from low to high. If `y` was 2-D, the coefficients for the data in column k of `y` are in column `k`. [residuals, rank, singular_values, rcond] : list These values are only returned if ``full == True`` - residuals -- sum of squared residuals of the least squares fit - rank -- the numerical rank of the scaled Vandermonde matrix - singular_values -- singular values of the scaled Vandermonde matrix - rcond -- value of `rcond`. For more details, see `numpy.linalg.lstsq`. Warns ----- RankWarning The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if ``full == False``. The warnings can be turned off by >>> import warnings >>> warnings.simplefilter('ignore', np.RankWarning) See Also -------- numpy.polynomial.polynomial.polyfit numpy.polynomial.legendre.legfit numpy.polynomial.laguerre.lagfit numpy.polynomial.hermite.hermfit numpy.polynomial.hermite_e.hermefit chebval : Evaluates a Chebyshev series. chebvander : Vandermonde matrix of Chebyshev series. chebweight : Chebyshev weight function. numpy.linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Chebyshev series `p` that minimizes the sum of the weighted squared errors .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, where :math:`w_j` are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation .. math:: V(x) * c = w * y, where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the coefficients to be solved for, `w` are the weights, and `y` are the observed values. This equation is then solved using the singular value decomposition of `V`. If some of the singular values of `V` are so small that they are neglected, then a `RankWarning` will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Chebyshev series are usually better conditioned than fits using power series, but much can depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", https://en.wikipedia.org/wiki/Curve_fitting Examples -------- |
169,792 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is a Chebyshev basis polynomial. This provides
better eigenvalue estimates than the unscaled case and for basis
polynomials the eigenvalues are guaranteed to be real if
`numpy.linalg.eigvalsh` is used to obtain them.
Parameters
----------
c : array_like
1-D array of Chebyshev series coefficients ordered from low to high
degree.
Returns
-------
mat : ndarray
Scaled companion matrix of dimensions (deg, deg).
Notes
-----
.. versionadded:: 1.7.0
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
raise ValueError('Series must have maximum degree of at least 1.')
if len(c) == 2:
return np.array([[-c[0]/c[1]]])
n = len(c) - 1
mat = np.zeros((n, n), dtype=c.dtype)
scl = np.array([1.] + [np.sqrt(.5)]*(n-1))
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[0] = np.sqrt(.5)
top[1:] = 1/2
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5
return mat
The provided code snippet includes necessary dependencies for implementing the `chebroots` function. Write a Python function `def chebroots(c)` to solve the following problem:
Compute the roots of a Chebyshev series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * T_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex. See Also -------- numpy.polynomial.polynomial.polyroots numpy.polynomial.legendre.legroots numpy.polynomial.laguerre.lagroots numpy.polynomial.hermite.hermroots numpy.polynomial.hermite_e.hermeroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method. The Chebyshev series basis polynomials aren't powers of `x` so the results of this function may seem unintuitive. Examples -------- >>> import numpy.polynomial.chebyshev as cheb >>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00]) # may vary
Here is the function:
def chebroots(c):
"""
Compute the roots of a Chebyshev series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * T_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
numpy.polynomial.polynomial.polyroots
numpy.polynomial.legendre.legroots
numpy.polynomial.laguerre.lagroots
numpy.polynomial.hermite.hermroots
numpy.polynomial.hermite_e.hermeroots
Notes
-----
The root estimates are obtained as the eigenvalues of the companion
matrix, Roots far from the origin of the complex plane may have large
errors due to the numerical instability of the series for such
values. Roots with multiplicity greater than 1 will also show larger
errors as the value of the series near such points is relatively
insensitive to errors in the roots. Isolated roots near the origin can
be improved by a few iterations of Newton's method.
The Chebyshev series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.chebyshev as cheb
>>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00]) # may vary
"""
# c is a trimmed copy
[c] = pu.as_series([c])
if len(c) < 2:
return np.array([], dtype=c.dtype)
if len(c) == 2:
return np.array([-c[0]/c[1]])
# rotated companion matrix reduces error
m = chebcompanion(c)[::-1,::-1]
r = la.eigvals(m)
r.sort()
return r | Compute the roots of a Chebyshev series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * T_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex. See Also -------- numpy.polynomial.polynomial.polyroots numpy.polynomial.legendre.legroots numpy.polynomial.laguerre.lagroots numpy.polynomial.hermite.hermroots numpy.polynomial.hermite_e.hermeroots Notes ----- The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton's method. The Chebyshev series basis polynomials aren't powers of `x` so the results of this function may seem unintuitive. Examples -------- >>> import numpy.polynomial.chebyshev as cheb >>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00]) # may vary |
169,793 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def chebvander(x, deg):
"""Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
`x`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., i] = T_i(x),
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the degree of the Chebyshev polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and
``chebval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of Chebyshev series of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray
The pseudo Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where The last index is the degree of the
corresponding Chebyshev polynomial. The dtype will be the same as
the converted `x`.
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
# Use forward recursion to generate the entries.
v[0] = x*0 + 1
if ideg > 0:
x2 = 2*x
v[1] = x
for i in range(2, ideg + 1):
v[i] = v[i-1]*x2 - v[i-2]
return np.moveaxis(v, 0, -1)
def chebpts1(npts):
"""
Chebyshev points of the first kind.
The Chebyshev points of the first kind are the points ``cos(x)``,
where ``x = [pi*(k + .5)/npts for k in range(npts)]``.
Parameters
----------
npts : int
Number of sample points desired.
Returns
-------
pts : ndarray
The Chebyshev points of the first kind.
See Also
--------
chebpts2
Notes
-----
.. versionadded:: 1.5.0
"""
_npts = int(npts)
if _npts != npts:
raise ValueError("npts must be integer")
if _npts < 1:
raise ValueError("npts must be >= 1")
x = 0.5 * np.pi / _npts * np.arange(-_npts+1, _npts+1, 2)
return np.sin(x)
The provided code snippet includes necessary dependencies for implementing the `chebinterpolate` function. Write a Python function `def chebinterpolate(func, deg, args=())` to solve the following problem:
Interpolate a function at the Chebyshev points of the first kind. Returns the Chebyshev series that interpolates `func` at the Chebyshev points of the first kind in the interval [-1, 1]. The interpolating series tends to a minmax approximation to `func` with increasing `deg` if the function is continuous in the interval. .. versionadded:: 1.14.0 Parameters ---------- func : function The function to be approximated. It must be a function of a single variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are extra arguments passed in the `args` parameter. deg : int Degree of the interpolating polynomial args : tuple, optional Extra arguments to be used in the function call. Default is no extra arguments. Returns ------- coef : ndarray, shape (deg + 1,) Chebyshev coefficients of the interpolating series ordered from low to high. Examples -------- >>> import numpy.polynomial.chebyshev as C >>> C.chebfromfunction(lambda x: np.tanh(x) + 0.5, 8) array([ 5.00000000e-01, 8.11675684e-01, -9.86864911e-17, -5.42457905e-02, -2.71387850e-16, 4.51658839e-03, 2.46716228e-17, -3.79694221e-04, -3.26899002e-16]) Notes ----- The Chebyshev polynomials used in the interpolation are orthogonal when sampled at the Chebyshev points of the first kind. If it is desired to constrain some of the coefficients they can simply be set to the desired value after the interpolation, no new interpolation or fit is needed. This is especially useful if it is known apriori that some of coefficients are zero. For instance, if the function is even then the coefficients of the terms of odd degree in the result can be set to zero.
Here is the function:
def chebinterpolate(func, deg, args=()):
"""Interpolate a function at the Chebyshev points of the first kind.
Returns the Chebyshev series that interpolates `func` at the Chebyshev
points of the first kind in the interval [-1, 1]. The interpolating
series tends to a minmax approximation to `func` with increasing `deg`
if the function is continuous in the interval.
.. versionadded:: 1.14.0
Parameters
----------
func : function
The function to be approximated. It must be a function of a single
variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are
extra arguments passed in the `args` parameter.
deg : int
Degree of the interpolating polynomial
args : tuple, optional
Extra arguments to be used in the function call. Default is no extra
arguments.
Returns
-------
coef : ndarray, shape (deg + 1,)
Chebyshev coefficients of the interpolating series ordered from low to
high.
Examples
--------
>>> import numpy.polynomial.chebyshev as C
>>> C.chebfromfunction(lambda x: np.tanh(x) + 0.5, 8)
array([ 5.00000000e-01, 8.11675684e-01, -9.86864911e-17,
-5.42457905e-02, -2.71387850e-16, 4.51658839e-03,
2.46716228e-17, -3.79694221e-04, -3.26899002e-16])
Notes
-----
The Chebyshev polynomials used in the interpolation are orthogonal when
sampled at the Chebyshev points of the first kind. If it is desired to
constrain some of the coefficients they can simply be set to the desired
value after the interpolation, no new interpolation or fit is needed. This
is especially useful if it is known apriori that some of coefficients are
zero. For instance, if the function is even then the coefficients of the
terms of odd degree in the result can be set to zero.
"""
deg = np.asarray(deg)
# check arguments.
if deg.ndim > 0 or deg.dtype.kind not in 'iu' or deg.size == 0:
raise TypeError("deg must be an int")
if deg < 0:
raise ValueError("expected deg >= 0")
order = deg + 1
xcheb = chebpts1(order)
yfunc = func(xcheb, *args)
m = chebvander(xcheb, deg)
c = np.dot(m.T, yfunc)
c[0] /= order
c[1:] /= 0.5*order
return c | Interpolate a function at the Chebyshev points of the first kind. Returns the Chebyshev series that interpolates `func` at the Chebyshev points of the first kind in the interval [-1, 1]. The interpolating series tends to a minmax approximation to `func` with increasing `deg` if the function is continuous in the interval. .. versionadded:: 1.14.0 Parameters ---------- func : function The function to be approximated. It must be a function of a single variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are extra arguments passed in the `args` parameter. deg : int Degree of the interpolating polynomial args : tuple, optional Extra arguments to be used in the function call. Default is no extra arguments. Returns ------- coef : ndarray, shape (deg + 1,) Chebyshev coefficients of the interpolating series ordered from low to high. Examples -------- >>> import numpy.polynomial.chebyshev as C >>> C.chebfromfunction(lambda x: np.tanh(x) + 0.5, 8) array([ 5.00000000e-01, 8.11675684e-01, -9.86864911e-17, -5.42457905e-02, -2.71387850e-16, 4.51658839e-03, 2.46716228e-17, -3.79694221e-04, -3.26899002e-16]) Notes ----- The Chebyshev polynomials used in the interpolation are orthogonal when sampled at the Chebyshev points of the first kind. If it is desired to constrain some of the coefficients they can simply be set to the desired value after the interpolation, no new interpolation or fit is needed. This is especially useful if it is known apriori that some of coefficients are zero. For instance, if the function is even then the coefficients of the terms of odd degree in the result can be set to zero. |
169,794 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `chebgauss` function. Write a Python function `def chebgauss(deg)` to solve the following problem:
Gauss-Chebyshev quadrature. Computes the sample points and weights for Gauss-Chebyshev quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with the weight function :math:`f(x) = 1/\\sqrt{1 - x^2}`. Parameters ---------- deg : int Number of sample points and weights. It must be >= 1. Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights. Notes ----- .. versionadded:: 1.7.0 The results have only been tested up to degree 100, higher degrees may be problematic. For Gauss-Chebyshev there are closed form solutions for the sample points and weights. If n = `deg`, then .. math:: x_i = \\cos(\\pi (2 i - 1) / (2 n)) .. math:: w_i = \\pi / n
Here is the function:
def chebgauss(deg):
"""
Gauss-Chebyshev quadrature.
Computes the sample points and weights for Gauss-Chebyshev quadrature.
These sample points and weights will correctly integrate polynomials of
degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
the weight function :math:`f(x) = 1/\\sqrt{1 - x^2}`.
Parameters
----------
deg : int
Number of sample points and weights. It must be >= 1.
Returns
-------
x : ndarray
1-D ndarray containing the sample points.
y : ndarray
1-D ndarray containing the weights.
Notes
-----
.. versionadded:: 1.7.0
The results have only been tested up to degree 100, higher degrees may
be problematic. For Gauss-Chebyshev there are closed form solutions for
the sample points and weights. If n = `deg`, then
.. math:: x_i = \\cos(\\pi (2 i - 1) / (2 n))
.. math:: w_i = \\pi / n
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg <= 0:
raise ValueError("deg must be a positive integer")
x = np.cos(np.pi * np.arange(1, 2*ideg, 2) / (2.0*ideg))
w = np.ones(ideg)*(np.pi/ideg)
return x, w | Gauss-Chebyshev quadrature. Computes the sample points and weights for Gauss-Chebyshev quadrature. These sample points and weights will correctly integrate polynomials of degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with the weight function :math:`f(x) = 1/\\sqrt{1 - x^2}`. Parameters ---------- deg : int Number of sample points and weights. It must be >= 1. Returns ------- x : ndarray 1-D ndarray containing the sample points. y : ndarray 1-D ndarray containing the weights. Notes ----- .. versionadded:: 1.7.0 The results have only been tested up to degree 100, higher degrees may be problematic. For Gauss-Chebyshev there are closed form solutions for the sample points and weights. If n = `deg`, then .. math:: x_i = \\cos(\\pi (2 i - 1) / (2 n)) .. math:: w_i = \\pi / n |
169,795 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `chebweight` function. Write a Python function `def chebweight(x)` to solve the following problem:
The weight function of the Chebyshev polynomials. The weight function is :math:`1/\\sqrt{1 - x^2}` and the interval of integration is :math:`[-1, 1]`. The Chebyshev polynomials are orthogonal, but not normalized, with respect to this weight function. Parameters ---------- x : array_like Values at which the weight function will be computed. Returns ------- w : ndarray The weight function at `x`. Notes ----- .. versionadded:: 1.7.0
Here is the function:
def chebweight(x):
"""
The weight function of the Chebyshev polynomials.
The weight function is :math:`1/\\sqrt{1 - x^2}` and the interval of
integration is :math:`[-1, 1]`. The Chebyshev polynomials are
orthogonal, but not normalized, with respect to this weight function.
Parameters
----------
x : array_like
Values at which the weight function will be computed.
Returns
-------
w : ndarray
The weight function at `x`.
Notes
-----
.. versionadded:: 1.7.0
"""
w = 1./(np.sqrt(1. + x) * np.sqrt(1. - x))
return w | The weight function of the Chebyshev polynomials. The weight function is :math:`1/\\sqrt{1 - x^2}` and the interval of integration is :math:`[-1, 1]`. The Chebyshev polynomials are orthogonal, but not normalized, with respect to this weight function. Parameters ---------- x : array_like Values at which the weight function will be computed. Returns ------- w : ndarray The weight function at `x`. Notes ----- .. versionadded:: 1.7.0 |
169,796 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `chebpts2` function. Write a Python function `def chebpts2(npts)` to solve the following problem:
Chebyshev points of the second kind. The Chebyshev points of the second kind are the points ``cos(x)``, where ``x = [pi*k/(npts - 1) for k in range(npts)]`` sorted in ascending order. Parameters ---------- npts : int Number of sample points desired. Returns ------- pts : ndarray The Chebyshev points of the second kind. Notes ----- .. versionadded:: 1.5.0
Here is the function:
def chebpts2(npts):
"""
Chebyshev points of the second kind.
The Chebyshev points of the second kind are the points ``cos(x)``,
where ``x = [pi*k/(npts - 1) for k in range(npts)]`` sorted in ascending
order.
Parameters
----------
npts : int
Number of sample points desired.
Returns
-------
pts : ndarray
The Chebyshev points of the second kind.
Notes
-----
.. versionadded:: 1.5.0
"""
_npts = int(npts)
if _npts != npts:
raise ValueError("npts must be integer")
if _npts < 2:
raise ValueError("npts must be >= 2")
x = np.linspace(-np.pi, 0, _npts)
return np.cos(x) | Chebyshev points of the second kind. The Chebyshev points of the second kind are the points ``cos(x)``, where ``x = [pi*k/(npts - 1) for k in range(npts)]`` sorted in ascending order. Parameters ---------- npts : int Number of sample points desired. Returns ------- pts : ndarray The Chebyshev points of the second kind. Notes ----- .. versionadded:: 1.5.0 |
169,797 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyline(off, scl):
"""
Returns an array representing a linear polynomial.
Parameters
----------
off, scl : scalars
The "y-intercept" and "slope" of the line, respectively.
Returns
-------
y : ndarray
This module's representation of the linear polynomial ``off +
scl*x``.
See Also
--------
numpy.polynomial.chebyshev.chebline
numpy.polynomial.legendre.legline
numpy.polynomial.laguerre.lagline
numpy.polynomial.hermite.hermline
numpy.polynomial.hermite_e.hermeline
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> P.polyline(1,-1)
array([ 1, -1])
>>> P.polyval(1, P.polyline(1,-1)) # should be 0
0.0
"""
if scl != 0:
return np.array([off, scl])
else:
return np.array([off])
def polymul(c1, c2):
"""
Multiply one polynomial by another.
Returns the product of two polynomials `c1` * `c2`. The arguments are
sequences of coefficients, from lowest order term to highest, e.g.,
[1,2,3] represents the polynomial ``1 + 2*x + 3*x**2.``
Parameters
----------
c1, c2 : array_like
1-D arrays of coefficients representing a polynomial, relative to the
"standard" basis, and ordered from lowest order term to highest.
Returns
-------
out : ndarray
Of the coefficients of their product.
See Also
--------
polyadd, polysub, polymulx, polydiv, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polymul(c1,c2)
array([ 3., 8., 14., 8., 3.])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
ret = np.convolve(c1, c2)
return pu.trimseq(ret)
The provided code snippet includes necessary dependencies for implementing the `polyfromroots` function. Write a Python function `def polyfromroots(roots)` to solve the following problem:
Generate a monic polynomial with given roots. Return the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), where the ``r_n`` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * x + ... + x^n The coefficient of the last term is 1 for monic polynomials in this form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of the polynomial's coefficients If all the roots are real, then `out` is also real, otherwise it is complex. (see Examples below). See Also -------- numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.laguerre.lagfromroots numpy.polynomial.hermite.hermfromroots numpy.polynomial.hermite_e.hermefromroots Notes ----- The coefficients are determined by multiplying together linear factors of the form ``(x - r_i)``, i.e. .. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n) where ``n == len(roots) - 1``; note that this implies that ``1`` is always returned for :math:`a_n`. Examples -------- >>> from numpy.polynomial import polynomial as P >>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x array([ 0., -1., 0., 1.]) >>> j = complex(0,1) >>> P.polyfromroots((-j,j)) # complex returned, though values are real array([1.+0.j, 0.+0.j, 1.+0.j])
Here is the function:
def polyfromroots(roots):
"""
Generate a monic polynomial with given roots.
Return the coefficients of the polynomial
.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
where the ``r_n`` are the roots specified in `roots`. If a zero has
multiplicity n, then it must appear in `roots` n times. For instance,
if 2 is a root of multiplicity three and 3 is a root of multiplicity 2,
then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear
in any order.
If the returned coefficients are `c`, then
.. math:: p(x) = c_0 + c_1 * x + ... + x^n
The coefficient of the last term is 1 for monic polynomials in this
form.
Parameters
----------
roots : array_like
Sequence containing the roots.
Returns
-------
out : ndarray
1-D array of the polynomial's coefficients If all the roots are
real, then `out` is also real, otherwise it is complex. (see
Examples below).
See Also
--------
numpy.polynomial.chebyshev.chebfromroots
numpy.polynomial.legendre.legfromroots
numpy.polynomial.laguerre.lagfromroots
numpy.polynomial.hermite.hermfromroots
numpy.polynomial.hermite_e.hermefromroots
Notes
-----
The coefficients are determined by multiplying together linear factors
of the form ``(x - r_i)``, i.e.
.. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n)
where ``n == len(roots) - 1``; note that this implies that ``1`` is always
returned for :math:`a_n`.
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x
array([ 0., -1., 0., 1.])
>>> j = complex(0,1)
>>> P.polyfromroots((-j,j)) # complex returned, though values are real
array([1.+0.j, 0.+0.j, 1.+0.j])
"""
return pu._fromroots(polyline, polymul, roots) | Generate a monic polynomial with given roots. Return the coefficients of the polynomial .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), where the ``r_n`` are the roots specified in `roots`. If a zero has multiplicity n, then it must appear in `roots` n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The roots can appear in any order. If the returned coefficients are `c`, then .. math:: p(x) = c_0 + c_1 * x + ... + x^n The coefficient of the last term is 1 for monic polynomials in this form. Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-D array of the polynomial's coefficients If all the roots are real, then `out` is also real, otherwise it is complex. (see Examples below). See Also -------- numpy.polynomial.chebyshev.chebfromroots numpy.polynomial.legendre.legfromroots numpy.polynomial.laguerre.lagfromroots numpy.polynomial.hermite.hermfromroots numpy.polynomial.hermite_e.hermefromroots Notes ----- The coefficients are determined by multiplying together linear factors of the form ``(x - r_i)``, i.e. .. math:: p(x) = (x - r_0) (x - r_1) ... (x - r_n) where ``n == len(roots) - 1``; note that this implies that ``1`` is always returned for :math:`a_n`. Examples -------- >>> from numpy.polynomial import polynomial as P >>> P.polyfromroots((-1,0,1)) # x(x - 1)(x + 1) = x^3 - x array([ 0., -1., 0., 1.]) >>> j = complex(0,1) >>> P.polyfromroots((-j,j)) # complex returned, though values are real array([1.+0.j, 0.+0.j, 1.+0.j]) |
169,798 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `polydiv` function. Write a Python function `def polydiv(c1, c2)` to solve the following problem:
Divide one polynomial by another. Returns the quotient-with-remainder of two polynomials `c1` / `c2`. The arguments are sequences of coefficients, from lowest order term to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``. Parameters ---------- c1, c2 : array_like 1-D arrays of polynomial coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of coefficient series representing the quotient and remainder. See Also -------- polyadd, polysub, polymulx, polymul, polypow Examples -------- >>> from numpy.polynomial import polynomial as P >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> P.polydiv(c1,c2) (array([3.]), array([-8., -4.])) >>> P.polydiv(c2,c1) (array([ 0.33333333]), array([ 2.66666667, 1.33333333])) # may vary
Here is the function:
def polydiv(c1, c2):
"""
Divide one polynomial by another.
Returns the quotient-with-remainder of two polynomials `c1` / `c2`.
The arguments are sequences of coefficients, from lowest order term
to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of polynomial coefficients ordered from low to high.
Returns
-------
[quo, rem] : ndarrays
Of coefficient series representing the quotient and remainder.
See Also
--------
polyadd, polysub, polymulx, polymul, polypow
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> P.polydiv(c1,c2)
(array([3.]), array([-8., -4.]))
>>> P.polydiv(c2,c1)
(array([ 0.33333333]), array([ 2.66666667, 1.33333333])) # may vary
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0:
raise ZeroDivisionError()
# note: this is more efficient than `pu._div(polymul, c1, c2)`
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2:
return c1[:1]*0, c1
elif lc2 == 1:
return c1/c2[-1], c1[:1]*0
else:
dlen = lc1 - lc2
scl = c2[-1]
c2 = c2[:-1]/scl
i = dlen
j = lc1 - 1
while i >= 0:
c1[i:j] -= c2*c1[j]
i -= 1
j -= 1
return c1[j+1:]/scl, pu.trimseq(c1[:j+1]) | Divide one polynomial by another. Returns the quotient-with-remainder of two polynomials `c1` / `c2`. The arguments are sequences of coefficients, from lowest order term to highest, e.g., [1,2,3] represents ``1 + 2*x + 3*x**2``. Parameters ---------- c1, c2 : array_like 1-D arrays of polynomial coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of coefficient series representing the quotient and remainder. See Also -------- polyadd, polysub, polymulx, polymul, polypow Examples -------- >>> from numpy.polynomial import polynomial as P >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> P.polydiv(c1,c2) (array([3.]), array([-8., -4.])) >>> P.polydiv(c2,c1) (array([ 0.33333333]), array([ 2.66666667, 1.33333333])) # may vary |
169,799 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `polypow` function. Write a Python function `def polypow(c, pow, maxpower=None)` to solve the following problem:
Raise a polynomial to a power. Returns the polynomial `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``1 + 2*x + 3*x**2.`` Parameters ---------- c : array_like 1-D array of array of series coefficients ordered from low to high degree. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Power series of power. See Also -------- polyadd, polysub, polymulx, polymul, polydiv Examples -------- >>> from numpy.polynomial import polynomial as P >>> P.polypow([1,2,3], 2) array([ 1., 4., 10., 12., 9.])
Here is the function:
def polypow(c, pow, maxpower=None):
"""Raise a polynomial to a power.
Returns the polynomial `c` raised to the power `pow`. The argument
`c` is a sequence of coefficients ordered from low to high. i.e.,
[1,2,3] is the series ``1 + 2*x + 3*x**2.``
Parameters
----------
c : array_like
1-D array of array of series coefficients ordered from low to
high degree.
pow : integer
Power to which the series will be raised
maxpower : integer, optional
Maximum power allowed. This is mainly to limit growth of the series
to unmanageable size. Default is 16
Returns
-------
coef : ndarray
Power series of power.
See Also
--------
polyadd, polysub, polymulx, polymul, polydiv
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> P.polypow([1,2,3], 2)
array([ 1., 4., 10., 12., 9.])
"""
# note: this is more efficient than `pu._pow(polymul, c1, c2)`, as it
# avoids calling `as_series` repeatedly
return pu._pow(np.convolve, c, pow, maxpower) | Raise a polynomial to a power. Returns the polynomial `c` raised to the power `pow`. The argument `c` is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series ``1 + 2*x + 3*x**2.`` Parameters ---------- c : array_like 1-D array of array of series coefficients ordered from low to high degree. pow : integer Power to which the series will be raised maxpower : integer, optional Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16 Returns ------- coef : ndarray Power series of power. See Also -------- polyadd, polysub, polymulx, polymul, polydiv Examples -------- >>> from numpy.polynomial import polynomial as P >>> P.polypow([1,2,3], 2) array([ 1., 4., 10., 12., 9.]) |
169,800 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `polyder` function. Write a Python function `def polyder(c, m=1, scl=1, axis=0)` to solve the following problem:
Differentiate a polynomial. Returns the polynomial coefficients `c` differentiated `m` times along `axis`. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of polynomial coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by `scl`. The end result is multiplication by ``scl**m``. This is for use in a linear change of variable. (Default: 1) axis : int, optional Axis over which the derivative is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- der : ndarray Polynomial coefficients of the derivative. See Also -------- polyint Examples -------- >>> from numpy.polynomial import polynomial as P >>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3 >>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2 array([ 2., 6., 12.]) >>> P.polyder(c,3) # (d**3/dx**3)(c) = 24 array([24.]) >>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2 array([ -2., -6., -12.]) >>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x array([ 6., 24.])
Here is the function:
def polyder(c, m=1, scl=1, axis=0):
"""
Differentiate a polynomial.
Returns the polynomial coefficients `c` differentiated `m` times along
`axis`. At each iteration the result is multiplied by `scl` (the
scaling factor is for use in a linear change of variable). The
argument `c` is an array of coefficients from low to high degree along
each axis, e.g., [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2``
while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is
``x`` and axis=1 is ``y``.
Parameters
----------
c : array_like
Array of polynomial coefficients. If c is multidimensional the
different axis correspond to different variables with the degree
in each axis given by the corresponding index.
m : int, optional
Number of derivatives taken, must be non-negative. (Default: 1)
scl : scalar, optional
Each differentiation is multiplied by `scl`. The end result is
multiplication by ``scl**m``. This is for use in a linear change
of variable. (Default: 1)
axis : int, optional
Axis over which the derivative is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
der : ndarray
Polynomial coefficients of the derivative.
See Also
--------
polyint
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3
>>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2
array([ 2., 6., 12.])
>>> P.polyder(c,3) # (d**3/dx**3)(c) = 24
array([24.])
>>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2
array([ -2., -6., -12.])
>>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x
array([ 6., 24.])
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
# astype fails with NA
c = c + 0.0
cdt = c.dtype
cnt = pu._deprecate_as_int(m, "the order of derivation")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of derivation must be non-negative")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
c = np.moveaxis(c, iaxis, 0)
n = len(c)
if cnt >= n:
c = c[:1]*0
else:
for i in range(cnt):
n = n - 1
c *= scl
der = np.empty((n,) + c.shape[1:], dtype=cdt)
for j in range(n, 0, -1):
der[j - 1] = j*c[j]
c = der
c = np.moveaxis(c, 0, iaxis)
return c | Differentiate a polynomial. Returns the polynomial coefficients `c` differentiated `m` times along `axis`. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of polynomial coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by `scl`. The end result is multiplication by ``scl**m``. This is for use in a linear change of variable. (Default: 1) axis : int, optional Axis over which the derivative is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- der : ndarray Polynomial coefficients of the derivative. See Also -------- polyint Examples -------- >>> from numpy.polynomial import polynomial as P >>> c = (1,2,3,4) # 1 + 2x + 3x**2 + 4x**3 >>> P.polyder(c) # (d/dx)(c) = 2 + 6x + 12x**2 array([ 2., 6., 12.]) >>> P.polyder(c,3) # (d**3/dx**3)(c) = 24 array([24.]) >>> P.polyder(c,scl=-1) # (d/d(-x))(c) = -2 - 6x - 12x**2 array([ -2., -6., -12.]) >>> P.polyder(c,2,-1) # (d**2/d(-x)**2)(c) = 6 + 24x array([ 6., 24.]) |
169,801 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyval(x, c, tensor=True):
"""
Evaluate a polynomial at points x.
If `c` is of length `n + 1`, this function returns the value
.. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
with themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, compatible object
The shape of the returned array is described above.
See Also
--------
polyval2d, polygrid2d, polyval3d, polygrid3d
Notes
-----
The evaluation uses Horner's method.
Examples
--------
>>> from numpy.polynomial.polynomial import polyval
>>> polyval(1, [1,2,3])
6.0
>>> a = np.arange(4).reshape(2,2)
>>> a
array([[0, 1],
[2, 3]])
>>> polyval(a, [1,2,3])
array([[ 1., 6.],
[17., 34.]])
>>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients
>>> coef
array([[0, 1],
[2, 3]])
>>> polyval([1,2], coef, tensor=True)
array([[2., 4.],
[4., 7.]])
>>> polyval([1,2], coef, tensor=False)
array([2., 7.])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
# astype fails with NA
c = c + 0.0
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
c0 = c[-1] + x*0
for i in range(2, len(c) + 1):
c0 = c[-i] + c0*x
return c0
The provided code snippet includes necessary dependencies for implementing the `polyint` function. Write a Python function `def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0)` to solve the following problem:
Integrate a polynomial. Returns the polynomial coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients, from low to high degree along each axis, e.g., [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like 1-D array of polynomial coefficients, ordered from low to high. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at zero is the first value in the list, the value of the second integral at zero is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray Coefficient array of the integral. Raises ------ ValueError If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- polyder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a` - perhaps not what one would have first thought. Examples -------- >>> from numpy.polynomial import polynomial as P >>> c = (1,2,3) >>> P.polyint(c) # should return array([0, 1, 1, 1]) array([0., 1., 1., 1.]) >>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20]) array([ 0. , 0. , 0. , 0.16666667, 0.08333333, # may vary 0.05 ]) >>> P.polyint(c,k=3) # should return array([3, 1, 1, 1]) array([3., 1., 1., 1.]) >>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1]) array([6., 1., 1., 1.]) >>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2]) array([ 0., -2., -2., -2.])
Here is the function:
def polyint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
"""
Integrate a polynomial.
Returns the polynomial coefficients `c` integrated `m` times from
`lbnd` along `axis`. At each iteration the resulting series is
**multiplied** by `scl` and an integration constant, `k`, is added.
The scaling factor is for use in a linear change of variable. ("Buyer
beware": note that, depending on what one is doing, one may want `scl`
to be the reciprocal of what one might expect; for more information,
see the Notes section below.) The argument `c` is an array of
coefficients, from low to high degree along each axis, e.g., [1,2,3]
represents the polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]]
represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is
``y``.
Parameters
----------
c : array_like
1-D array of polynomial coefficients, ordered from low to high.
m : int, optional
Order of integration, must be positive. (Default: 1)
k : {[], list, scalar}, optional
Integration constant(s). The value of the first integral at zero
is the first value in the list, the value of the second integral
at zero is the second value, etc. If ``k == []`` (the default),
all constants are set to zero. If ``m == 1``, a single scalar can
be given instead of a list.
lbnd : scalar, optional
The lower bound of the integral. (Default: 0)
scl : scalar, optional
Following each integration the result is *multiplied* by `scl`
before the integration constant is added. (Default: 1)
axis : int, optional
Axis over which the integral is taken. (Default: 0).
.. versionadded:: 1.7.0
Returns
-------
S : ndarray
Coefficient array of the integral.
Raises
------
ValueError
If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
``np.ndim(scl) != 0``.
See Also
--------
polyder
Notes
-----
Note that the result of each integration is *multiplied* by `scl`. Why
is this important to note? Say one is making a linear change of
variable :math:`u = ax + b` in an integral relative to `x`. Then
:math:`dx = du/a`, so one will need to set `scl` equal to
:math:`1/a` - perhaps not what one would have first thought.
Examples
--------
>>> from numpy.polynomial import polynomial as P
>>> c = (1,2,3)
>>> P.polyint(c) # should return array([0, 1, 1, 1])
array([0., 1., 1., 1.])
>>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20])
array([ 0. , 0. , 0. , 0.16666667, 0.08333333, # may vary
0.05 ])
>>> P.polyint(c,k=3) # should return array([3, 1, 1, 1])
array([3., 1., 1., 1.])
>>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1])
array([6., 1., 1., 1.])
>>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2])
array([ 0., -2., -2., -2.])
"""
c = np.array(c, ndmin=1, copy=True)
if c.dtype.char in '?bBhHiIlLqQpP':
# astype doesn't preserve mask attribute.
c = c + 0.0
cdt = c.dtype
if not np.iterable(k):
k = [k]
cnt = pu._deprecate_as_int(m, "the order of integration")
iaxis = pu._deprecate_as_int(axis, "the axis")
if cnt < 0:
raise ValueError("The order of integration must be non-negative")
if len(k) > cnt:
raise ValueError("Too many integration constants")
if np.ndim(lbnd) != 0:
raise ValueError("lbnd must be a scalar.")
if np.ndim(scl) != 0:
raise ValueError("scl must be a scalar.")
iaxis = normalize_axis_index(iaxis, c.ndim)
if cnt == 0:
return c
k = list(k) + [0]*(cnt - len(k))
c = np.moveaxis(c, iaxis, 0)
for i in range(cnt):
n = len(c)
c *= scl
if n == 1 and np.all(c[0] == 0):
c[0] += k[i]
else:
tmp = np.empty((n + 1,) + c.shape[1:], dtype=cdt)
tmp[0] = c[0]*0
tmp[1] = c[0]
for j in range(1, n):
tmp[j + 1] = c[j]/(j + 1)
tmp[0] += k[i] - polyval(lbnd, tmp)
c = tmp
c = np.moveaxis(c, 0, iaxis)
return c | Integrate a polynomial. Returns the polynomial coefficients `c` integrated `m` times from `lbnd` along `axis`. At each iteration the resulting series is **multiplied** by `scl` and an integration constant, `k`, is added. The scaling factor is for use in a linear change of variable. ("Buyer beware": note that, depending on what one is doing, one may want `scl` to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument `c` is an array of coefficients, from low to high degree along each axis, e.g., [1,2,3] represents the polynomial ``1 + 2*x + 3*x**2`` while [[1,2],[1,2]] represents ``1 + 1*x + 2*y + 2*x*y`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like 1-D array of polynomial coefficients, ordered from low to high. m : int, optional Order of integration, must be positive. (Default: 1) k : {[], list, scalar}, optional Integration constant(s). The value of the first integral at zero is the first value in the list, the value of the second integral at zero is the second value, etc. If ``k == []`` (the default), all constants are set to zero. If ``m == 1``, a single scalar can be given instead of a list. lbnd : scalar, optional The lower bound of the integral. (Default: 0) scl : scalar, optional Following each integration the result is *multiplied* by `scl` before the integration constant is added. (Default: 1) axis : int, optional Axis over which the integral is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- S : ndarray Coefficient array of the integral. Raises ------ ValueError If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or ``np.ndim(scl) != 0``. See Also -------- polyder Notes ----- Note that the result of each integration is *multiplied* by `scl`. Why is this important to note? Say one is making a linear change of variable :math:`u = ax + b` in an integral relative to `x`. Then :math:`dx = du/a`, so one will need to set `scl` equal to :math:`1/a` - perhaps not what one would have first thought. Examples -------- >>> from numpy.polynomial import polynomial as P >>> c = (1,2,3) >>> P.polyint(c) # should return array([0, 1, 1, 1]) array([0., 1., 1., 1.]) >>> P.polyint(c,3) # should return array([0, 0, 0, 1/6, 1/12, 1/20]) array([ 0. , 0. , 0. , 0.16666667, 0.08333333, # may vary 0.05 ]) >>> P.polyint(c,k=3) # should return array([3, 1, 1, 1]) array([3., 1., 1., 1.]) >>> P.polyint(c,lbnd=-2) # should return array([6, 1, 1, 1]) array([6., 1., 1., 1.]) >>> P.polyint(c,scl=-2) # should return array([0, -2, -2, -2]) array([ 0., -2., -2., -2.]) |
169,802 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
The provided code snippet includes necessary dependencies for implementing the `polyvalfromroots` function. Write a Python function `def polyvalfromroots(x, r, tensor=True)` to solve the following problem:
Evaluate a polynomial specified by its roots at points x. If `r` is of length `N`, this function returns the value .. math:: p(x) = \\prod_{n=1}^{N} (x - r_n) The parameter `x` is converted to an array only if it is a tuple or a list, otherwise it is treated as a scalar. In either case, either `x` or its elements must support multiplication and addition both with themselves and with the elements of `r`. If `r` is a 1-D array, then `p(x)` will have the same shape as `x`. If `r` is multidimensional, then the shape of the result depends on the value of `tensor`. If `tensor` is ``True`` the shape will be r.shape[1:] + x.shape; that is, each polynomial is evaluated at every value of `x`. If `tensor` is ``False``, the shape will be r.shape[1:]; that is, each polynomial is evaluated only for the corresponding broadcast value of `x`. Note that scalars have shape (,). .. versionadded:: 1.12 Parameters ---------- x : array_like, compatible object If `x` is a list or tuple, it is converted to an ndarray, otherwise it is left unchanged and treated as a scalar. In either case, `x` or its elements must support addition and multiplication with with themselves and with the elements of `r`. r : array_like Array of roots. If `r` is multidimensional the first index is the root index, while the remaining indices enumerate multiple polynomials. For instance, in the two dimensional case the roots of each polynomial may be thought of as stored in the columns of `r`. tensor : boolean, optional If True, the shape of the roots array is extended with ones on the right, one for each dimension of `x`. Scalars have dimension 0 for this action. The result is that every column of coefficients in `r` is evaluated for every element of `x`. If False, `x` is broadcast over the columns of `r` for the evaluation. This keyword is useful when `r` is multidimensional. The default value is True. Returns ------- values : ndarray, compatible object The shape of the returned array is described above. See Also -------- polyroots, polyfromroots, polyval Examples -------- >>> from numpy.polynomial.polynomial import polyvalfromroots >>> polyvalfromroots(1, [1,2,3]) 0.0 >>> a = np.arange(4).reshape(2,2) >>> a array([[0, 1], [2, 3]]) >>> polyvalfromroots(a, [-1, 0, 1]) array([[-0., 0.], [ 6., 24.]]) >>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients >>> r # each column of r defines one polynomial array([[-2, -1], [ 0, 1]]) >>> b = [-2, 1] >>> polyvalfromroots(b, r, tensor=True) array([[-0., 3.], [ 3., 0.]]) >>> polyvalfromroots(b, r, tensor=False) array([-0., 0.])
Here is the function:
def polyvalfromroots(x, r, tensor=True):
"""
Evaluate a polynomial specified by its roots at points x.
If `r` is of length `N`, this function returns the value
.. math:: p(x) = \\prod_{n=1}^{N} (x - r_n)
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `r`.
If `r` is a 1-D array, then `p(x)` will have the same shape as `x`. If `r`
is multidimensional, then the shape of the result depends on the value of
`tensor`. If `tensor` is ``True`` the shape will be r.shape[1:] + x.shape;
that is, each polynomial is evaluated at every value of `x`. If `tensor` is
``False``, the shape will be r.shape[1:]; that is, each polynomial is
evaluated only for the corresponding broadcast value of `x`. Note that
scalars have shape (,).
.. versionadded:: 1.12
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
with themselves and with the elements of `r`.
r : array_like
Array of roots. If `r` is multidimensional the first index is the
root index, while the remaining indices enumerate multiple
polynomials. For instance, in the two dimensional case the roots
of each polynomial may be thought of as stored in the columns of `r`.
tensor : boolean, optional
If True, the shape of the roots array is extended with ones on the
right, one for each dimension of `x`. Scalars have dimension 0 for this
action. The result is that every column of coefficients in `r` is
evaluated for every element of `x`. If False, `x` is broadcast over the
columns of `r` for the evaluation. This keyword is useful when `r` is
multidimensional. The default value is True.
Returns
-------
values : ndarray, compatible object
The shape of the returned array is described above.
See Also
--------
polyroots, polyfromroots, polyval
Examples
--------
>>> from numpy.polynomial.polynomial import polyvalfromroots
>>> polyvalfromroots(1, [1,2,3])
0.0
>>> a = np.arange(4).reshape(2,2)
>>> a
array([[0, 1],
[2, 3]])
>>> polyvalfromroots(a, [-1, 0, 1])
array([[-0., 0.],
[ 6., 24.]])
>>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients
>>> r # each column of r defines one polynomial
array([[-2, -1],
[ 0, 1]])
>>> b = [-2, 1]
>>> polyvalfromroots(b, r, tensor=True)
array([[-0., 3.],
[ 3., 0.]])
>>> polyvalfromroots(b, r, tensor=False)
array([-0., 0.])
"""
r = np.array(r, ndmin=1, copy=False)
if r.dtype.char in '?bBhHiIlLqQpP':
r = r.astype(np.double)
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray):
if tensor:
r = r.reshape(r.shape + (1,)*x.ndim)
elif x.ndim >= r.ndim:
raise ValueError("x.ndim must be < r.ndim when tensor == False")
return np.prod(x - r, axis=0) | Evaluate a polynomial specified by its roots at points x. If `r` is of length `N`, this function returns the value .. math:: p(x) = \\prod_{n=1}^{N} (x - r_n) The parameter `x` is converted to an array only if it is a tuple or a list, otherwise it is treated as a scalar. In either case, either `x` or its elements must support multiplication and addition both with themselves and with the elements of `r`. If `r` is a 1-D array, then `p(x)` will have the same shape as `x`. If `r` is multidimensional, then the shape of the result depends on the value of `tensor`. If `tensor` is ``True`` the shape will be r.shape[1:] + x.shape; that is, each polynomial is evaluated at every value of `x`. If `tensor` is ``False``, the shape will be r.shape[1:]; that is, each polynomial is evaluated only for the corresponding broadcast value of `x`. Note that scalars have shape (,). .. versionadded:: 1.12 Parameters ---------- x : array_like, compatible object If `x` is a list or tuple, it is converted to an ndarray, otherwise it is left unchanged and treated as a scalar. In either case, `x` or its elements must support addition and multiplication with with themselves and with the elements of `r`. r : array_like Array of roots. If `r` is multidimensional the first index is the root index, while the remaining indices enumerate multiple polynomials. For instance, in the two dimensional case the roots of each polynomial may be thought of as stored in the columns of `r`. tensor : boolean, optional If True, the shape of the roots array is extended with ones on the right, one for each dimension of `x`. Scalars have dimension 0 for this action. The result is that every column of coefficients in `r` is evaluated for every element of `x`. If False, `x` is broadcast over the columns of `r` for the evaluation. This keyword is useful when `r` is multidimensional. The default value is True. Returns ------- values : ndarray, compatible object The shape of the returned array is described above. See Also -------- polyroots, polyfromroots, polyval Examples -------- >>> from numpy.polynomial.polynomial import polyvalfromroots >>> polyvalfromroots(1, [1,2,3]) 0.0 >>> a = np.arange(4).reshape(2,2) >>> a array([[0, 1], [2, 3]]) >>> polyvalfromroots(a, [-1, 0, 1]) array([[-0., 0.], [ 6., 24.]]) >>> r = np.arange(-2, 2).reshape(2,2) # multidimensional coefficients >>> r # each column of r defines one polynomial array([[-2, -1], [ 0, 1]]) >>> b = [-2, 1] >>> polyvalfromroots(b, r, tensor=True) array([[-0., 3.], [ 3., 0.]]) >>> polyvalfromroots(b, r, tensor=False) array([-0., 0.]) |
169,803 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyval(x, c, tensor=True):
"""
Evaluate a polynomial at points x.
If `c` is of length `n + 1`, this function returns the value
.. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
with themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, compatible object
The shape of the returned array is described above.
See Also
--------
polyval2d, polygrid2d, polyval3d, polygrid3d
Notes
-----
The evaluation uses Horner's method.
Examples
--------
>>> from numpy.polynomial.polynomial import polyval
>>> polyval(1, [1,2,3])
6.0
>>> a = np.arange(4).reshape(2,2)
>>> a
array([[0, 1],
[2, 3]])
>>> polyval(a, [1,2,3])
array([[ 1., 6.],
[17., 34.]])
>>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients
>>> coef
array([[0, 1],
[2, 3]])
>>> polyval([1,2], coef, tensor=True)
array([[2., 4.],
[4., 7.]])
>>> polyval([1,2], coef, tensor=False)
array([2., 7.])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
# astype fails with NA
c = c + 0.0
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
c0 = c[-1] + x*0
for i in range(2, len(c) + 1):
c0 = c[-i] + c0*x
return c0
The provided code snippet includes necessary dependencies for implementing the `polyval2d` function. Write a Python function `def polyval2d(x, y, c)` to solve the following problem:
Evaluate a 2-D polynomial at points (x, y). This function returns the value .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points formed with pairs of corresponding values from `x` and `y`. See Also -------- polyval, polygrid2d, polyval3d, polygrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def polyval2d(x, y, c):
"""
Evaluate a 2-D polynomial at points (x, y).
This function returns the value
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars and they
must have the same shape after conversion. In either case, either `x`
and `y` or their elements must support multiplication and addition both
with themselves and with the elements of `c`.
If `c` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points `(x, y)`,
where `x` and `y` must have the same shape. If `x` or `y` is a list
or tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term
of multi-degree i,j is contained in `c[i,j]`. If `c` has
dimension greater than two the remaining indices enumerate multiple
sets of coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points formed with
pairs of corresponding values from `x` and `y`.
See Also
--------
polyval, polygrid2d, polyval3d, polygrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(polyval, c, x, y) | Evaluate a 2-D polynomial at points (x, y). This function returns the value .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * x^i * y^j The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points `(x, y)`, where `x` and `y` must have the same shape. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in `c[i,j]`. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points formed with pairs of corresponding values from `x` and `y`. See Also -------- polyval, polygrid2d, polyval3d, polygrid3d Notes ----- .. versionadded:: 1.7.0 |
169,804 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyval(x, c, tensor=True):
"""
Evaluate a polynomial at points x.
If `c` is of length `n + 1`, this function returns the value
.. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
with themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, compatible object
The shape of the returned array is described above.
See Also
--------
polyval2d, polygrid2d, polyval3d, polygrid3d
Notes
-----
The evaluation uses Horner's method.
Examples
--------
>>> from numpy.polynomial.polynomial import polyval
>>> polyval(1, [1,2,3])
6.0
>>> a = np.arange(4).reshape(2,2)
>>> a
array([[0, 1],
[2, 3]])
>>> polyval(a, [1,2,3])
array([[ 1., 6.],
[17., 34.]])
>>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients
>>> coef
array([[0, 1],
[2, 3]])
>>> polyval([1,2], coef, tensor=True)
array([[2., 4.],
[4., 7.]])
>>> polyval([1,2], coef, tensor=False)
array([2., 7.])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
# astype fails with NA
c = c + 0.0
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
c0 = c[-1] + x*0
for i in range(2, len(c) + 1):
c0 = c[-i] + c0*x
return c0
The provided code snippet includes necessary dependencies for implementing the `polygrid2d` function. Write a Python function `def polygrid2d(x, y, c)` to solve the following problem:
Evaluate a 2-D polynomial on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * a^i * b^j where the points `(a, b)` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- polyval, polyval2d, polyval3d, polygrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def polygrid2d(x, y, c):
"""
Evaluate a 2-D polynomial on the Cartesian product of x and y.
This function returns the values:
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * a^i * b^j
where the points `(a, b)` consist of all pairs formed by taking
`a` from `x` and `b` from `y`. The resulting points form a grid with
`x` in the first dimension and `y` in the second.
The parameters `x` and `y` are converted to arrays only if they are
tuples or a lists, otherwise they are treated as a scalars. In either
case, either `x` and `y` or their elements must support multiplication
and addition both with themselves and with the elements of `c`.
If `c` has fewer than two dimensions, ones are implicitly appended to
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
x.shape + y.shape.
Parameters
----------
x, y : array_like, compatible objects
The two dimensional series is evaluated at the points in the
Cartesian product of `x` and `y`. If `x` or `y` is a list or
tuple, it is first converted to an ndarray, otherwise it is left
unchanged and, if it isn't an ndarray, it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree i,j are contained in ``c[i,j]``. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points in the Cartesian
product of `x` and `y`.
See Also
--------
polyval, polyval2d, polyval3d, polygrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(polyval, c, x, y) | Evaluate a 2-D polynomial on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * a^i * b^j where the points `(a, b)` consist of all pairs formed by taking `a` from `x` and `b` from `y`. The resulting points form a grid with `x` in the first dimension and `y` in the second. The parameters `x` and `y` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x` and `y` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape. Parameters ---------- x, y : array_like, compatible objects The two dimensional series is evaluated at the points in the Cartesian product of `x` and `y`. If `x` or `y` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- polyval, polyval2d, polyval3d, polygrid3d Notes ----- .. versionadded:: 1.7.0 |
169,805 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyval(x, c, tensor=True):
"""
Evaluate a polynomial at points x.
If `c` is of length `n + 1`, this function returns the value
.. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
with themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, compatible object
The shape of the returned array is described above.
See Also
--------
polyval2d, polygrid2d, polyval3d, polygrid3d
Notes
-----
The evaluation uses Horner's method.
Examples
--------
>>> from numpy.polynomial.polynomial import polyval
>>> polyval(1, [1,2,3])
6.0
>>> a = np.arange(4).reshape(2,2)
>>> a
array([[0, 1],
[2, 3]])
>>> polyval(a, [1,2,3])
array([[ 1., 6.],
[17., 34.]])
>>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients
>>> coef
array([[0, 1],
[2, 3]])
>>> polyval([1,2], coef, tensor=True)
array([[2., 4.],
[4., 7.]])
>>> polyval([1,2], coef, tensor=False)
array([2., 7.])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
# astype fails with NA
c = c + 0.0
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
c0 = c[-1] + x*0
for i in range(2, len(c) + 1):
c0 = c[-i] + c0*x
return c0
The provided code snippet includes necessary dependencies for implementing the `polyval3d` function. Write a Python function `def polyval3d(x, y, z, c)` to solve the following problem:
Evaluate a 3-D polynomial at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- polyval, polyval2d, polygrid2d, polygrid3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def polyval3d(x, y, z, c):
"""
Evaluate a 3-D polynomial at points (x, y, z).
This function returns the values:
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k
The parameters `x`, `y`, and `z` are converted to arrays only if
they are tuples or a lists, otherwise they are treated as a scalars and
they must have the same shape after conversion. In either case, either
`x`, `y`, and `z` or their elements must support multiplication and
addition both with themselves and with the elements of `c`.
If `c` has fewer than 3 dimensions, ones are implicitly appended to its
shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape.
Parameters
----------
x, y, z : array_like, compatible object
The three dimensional series is evaluated at the points
`(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If
any of `x`, `y`, or `z` is a list or tuple, it is first converted
to an ndarray, otherwise it is left unchanged and if it isn't an
ndarray it is treated as a scalar.
c : array_like
Array of coefficients ordered so that the coefficient of the term of
multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
greater than 3 the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the multidimensional polynomial on points formed with
triples of corresponding values from `x`, `y`, and `z`.
See Also
--------
polyval, polyval2d, polygrid2d, polygrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(polyval, c, x, y, z) | Evaluate a 3-D polynomial at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * x^i * y^j * z^k The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. Parameters ---------- x, y, z : array_like, compatible object The three dimensional series is evaluated at the points `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If any of `x`, `y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ndarray it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the multidimensional polynomial on points formed with triples of corresponding values from `x`, `y`, and `z`. See Also -------- polyval, polyval2d, polygrid2d, polygrid3d Notes ----- .. versionadded:: 1.7.0 |
169,806 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyval(x, c, tensor=True):
"""
Evaluate a polynomial at points x.
If `c` is of length `n + 1`, this function returns the value
.. math:: p(x) = c_0 + c_1 * x + ... + c_n * x^n
The parameter `x` is converted to an array only if it is a tuple or a
list, otherwise it is treated as a scalar. In either case, either `x`
or its elements must support multiplication and addition both with
themselves and with the elements of `c`.
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
`c` is multidimensional, then the shape of the result depends on the
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so
they should be avoided if efficiency is a concern.
Parameters
----------
x : array_like, compatible object
If `x` is a list or tuple, it is converted to an ndarray, otherwise
it is left unchanged and treated as a scalar. In either case, `x`
or its elements must support addition and multiplication with
with themselves and with the elements of `c`.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree n are contained in c[n]. If `c` is multidimensional the
remaining indices enumerate multiple polynomials. In the two
dimensional case the coefficients may be thought of as stored in
the columns of `c`.
tensor : boolean, optional
If True, the shape of the coefficient array is extended with ones
on the right, one for each dimension of `x`. Scalars have dimension 0
for this action. The result is that every column of coefficients in
`c` is evaluated for every element of `x`. If False, `x` is broadcast
over the columns of `c` for the evaluation. This keyword is useful
when `c` is multidimensional. The default value is True.
.. versionadded:: 1.7.0
Returns
-------
values : ndarray, compatible object
The shape of the returned array is described above.
See Also
--------
polyval2d, polygrid2d, polyval3d, polygrid3d
Notes
-----
The evaluation uses Horner's method.
Examples
--------
>>> from numpy.polynomial.polynomial import polyval
>>> polyval(1, [1,2,3])
6.0
>>> a = np.arange(4).reshape(2,2)
>>> a
array([[0, 1],
[2, 3]])
>>> polyval(a, [1,2,3])
array([[ 1., 6.],
[17., 34.]])
>>> coef = np.arange(4).reshape(2,2) # multidimensional coefficients
>>> coef
array([[0, 1],
[2, 3]])
>>> polyval([1,2], coef, tensor=True)
array([[2., 4.],
[4., 7.]])
>>> polyval([1,2], coef, tensor=False)
array([2., 7.])
"""
c = np.array(c, ndmin=1, copy=False)
if c.dtype.char in '?bBhHiIlLqQpP':
# astype fails with NA
c = c + 0.0
if isinstance(x, (tuple, list)):
x = np.asarray(x)
if isinstance(x, np.ndarray) and tensor:
c = c.reshape(c.shape + (1,)*x.ndim)
c0 = c[-1] + x*0
for i in range(2, len(c) + 1):
c0 = c[-i] + c0*x
return c0
The provided code snippet includes necessary dependencies for implementing the `polygrid3d` function. Write a Python function `def polygrid3d(x, y, z, c)` to solve the following problem:
Evaluate a 3-D polynomial on the Cartesian product of x, y and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * a^i * b^j * c^k where the points `(a, b, c)` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- polyval, polyval2d, polygrid2d, polyval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def polygrid3d(x, y, z, c):
"""
Evaluate a 3-D polynomial on the Cartesian product of x, y and z.
This function returns the values:
.. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * a^i * b^j * c^k
where the points `(a, b, c)` consist of all triples formed by taking
`a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
a grid with `x` in the first dimension, `y` in the second, and `z` in
the third.
The parameters `x`, `y`, and `z` are converted to arrays only if they
are tuples or a lists, otherwise they are treated as a scalars. In
either case, either `x`, `y`, and `z` or their elements must support
multiplication and addition both with themselves and with the elements
of `c`.
If `c` has fewer than three dimensions, ones are implicitly appended to
its shape to make it 3-D. The shape of the result will be c.shape[3:] +
x.shape + y.shape + z.shape.
Parameters
----------
x, y, z : array_like, compatible objects
The three dimensional series is evaluated at the points in the
Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a
list or tuple, it is first converted to an ndarray, otherwise it is
left unchanged and, if it isn't an ndarray, it is treated as a
scalar.
c : array_like
Array of coefficients ordered so that the coefficients for terms of
degree i,j are contained in ``c[i,j]``. If `c` has dimension
greater than two the remaining indices enumerate multiple sets of
coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional polynomial at points in the Cartesian
product of `x` and `y`.
See Also
--------
polyval, polyval2d, polygrid2d, polyval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(polyval, c, x, y, z) | Evaluate a 3-D polynomial on the Cartesian product of x, y and z. This function returns the values: .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * a^i * b^j * c^k where the points `(a, b, c)` consist of all triples formed by taking `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form a grid with `x` in the first dimension, `y` in the second, and `z` in the third. The parameters `x`, `y`, and `z` are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either `x`, `y`, and `z` or their elements must support multiplication and addition both with themselves and with the elements of `c`. If `c` has fewer than three dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape + y.shape + z.shape. Parameters ---------- x, y, z : array_like, compatible objects The three dimensional series is evaluated at the points in the Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, if it isn't an ndarray, it is treated as a scalar. c : array_like Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in ``c[i,j]``. If `c` has dimension greater than two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional polynomial at points in the Cartesian product of `x` and `y`. See Also -------- polyval, polyval2d, polygrid2d, polyval3d Notes ----- .. versionadded:: 1.7.0 |
169,807 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyvander(x, deg):
"""Vandermonde matrix of given degree.
Returns the Vandermonde matrix of degree `deg` and sample points
`x`. The Vandermonde matrix is defined by
.. math:: V[..., i] = x^i,
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the power of `x`.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
matrix ``V = polyvander(x, n)``, then ``np.dot(V, c)`` and
``polyval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of polynomials of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray.
The Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where the last index is the power of `x`.
The dtype will be the same as the converted `x`.
See Also
--------
polyvander2d, polyvander3d
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
if ideg > 0:
v[1] = x
for i in range(2, ideg + 1):
v[i] = v[i-1]*x
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `polyvander2d` function. Write a Python function `def polyvander2d(x, y, deg)` to solve the following problem:
Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y)`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = x^i * y^j, where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of `V` index the points `(x, y)` and the last index encodes the powers of `x` and `y`. If ``V = polyvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``polyval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D polynomials of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg([1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- polyvander, polyvander3d, polyval2d, polyval3d
Here is the function:
def polyvander2d(x, y, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y)`. The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (deg[1] + 1)*i + j] = x^i * y^j,
where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
`V` index the points `(x, y)` and the last index encodes the powers of
`x` and `y`.
If ``V = polyvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
correspond to the elements of a 2-D coefficient array `c` of shape
(xdeg + 1, ydeg + 1) in the order
.. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
and ``np.dot(V, c.flat)`` and ``polyval2d(x, y, c)`` will be the same
up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 2-D polynomials
of the same degrees and sample points.
Parameters
----------
x, y : array_like
Arrays of point coordinates, all of the same shape. The dtypes
will be converted to either float64 or complex128 depending on
whether any of the elements are complex. Scalars are converted to
1-D arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg].
Returns
-------
vander2d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg([1]+1)`. The dtype will be the same
as the converted `x` and `y`.
See Also
--------
polyvander, polyvander3d, polyval2d, polyval3d
"""
return pu._vander_nd_flat((polyvander, polyvander), (x, y), deg) | Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y)`. The pseudo-Vandermonde matrix is defined by .. math:: V[..., (deg[1] + 1)*i + j] = x^i * y^j, where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of `V` index the points `(x, y)` and the last index encodes the powers of `x` and `y`. If ``V = polyvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` correspond to the elements of a 2-D coefficient array `c` of shape (xdeg + 1, ydeg + 1) in the order .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... and ``np.dot(V, c.flat)`` and ``polyval2d(x, y, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D polynomials of the same degrees and sample points. Parameters ---------- x, y : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg]. Returns ------- vander2d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg([1]+1)`. The dtype will be the same as the converted `x` and `y`. See Also -------- polyvander, polyvander3d, polyval2d, polyval3d |
169,808 | import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
def polyvander(x, deg):
"""Vandermonde matrix of given degree.
Returns the Vandermonde matrix of degree `deg` and sample points
`x`. The Vandermonde matrix is defined by
.. math:: V[..., i] = x^i,
where `0 <= i <= deg`. The leading indices of `V` index the elements of
`x` and the last index is the power of `x`.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
matrix ``V = polyvander(x, n)``, then ``np.dot(V, c)`` and
``polyval(x, c)`` are the same up to roundoff. This equivalence is
useful both for least squares fitting and for the evaluation of a large
number of polynomials of the same degree and sample points.
Parameters
----------
x : array_like
Array of points. The dtype is converted to float64 or complex128
depending on whether any of the elements are complex. If `x` is
scalar it is converted to a 1-D array.
deg : int
Degree of the resulting matrix.
Returns
-------
vander : ndarray.
The Vandermonde matrix. The shape of the returned matrix is
``x.shape + (deg + 1,)``, where the last index is the power of `x`.
The dtype will be the same as the converted `x`.
See Also
--------
polyvander2d, polyvander3d
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg < 0:
raise ValueError("deg must be non-negative")
x = np.array(x, copy=False, ndmin=1) + 0.0
dims = (ideg + 1,) + x.shape
dtyp = x.dtype
v = np.empty(dims, dtype=dtyp)
v[0] = x*0 + 1
if ideg > 0:
v[1] = x
for i in range(2, ideg + 1):
v[i] = v[i-1]*x
return np.moveaxis(v, 0, -1)
The provided code snippet includes necessary dependencies for implementing the `polyvander3d` function. Write a Python function `def polyvander3d(x, y, z, deg)` to solve the following problem:
Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = x^i * y^j * z^k, where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading indices of `V` index the points `(x, y, z)` and the last index encodes the powers of `x`, `y`, and `z`. If ``V = polyvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``polyval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D polynomials of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- polyvander, polyvander3d, polyval2d, polyval3d Notes ----- .. versionadded:: 1.7.0
Here is the function:
def polyvander3d(x, y, z, deg):
"""Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
then The pseudo-Vandermonde matrix is defined by
.. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = x^i * y^j * z^k,
where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading
indices of `V` index the points `(x, y, z)` and the last index encodes
the powers of `x`, `y`, and `z`.
If ``V = polyvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
of `V` correspond to the elements of a 3-D coefficient array `c` of
shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
.. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
and ``np.dot(V, c.flat)`` and ``polyval3d(x, y, z, c)`` will be the
same up to roundoff. This equivalence is useful both for least squares
fitting and for the evaluation of a large number of 3-D polynomials
of the same degrees and sample points.
Parameters
----------
x, y, z : array_like
Arrays of point coordinates, all of the same shape. The dtypes will
be converted to either float64 or complex128 depending on whether
any of the elements are complex. Scalars are converted to 1-D
arrays.
deg : list of ints
List of maximum degrees of the form [x_deg, y_deg, z_deg].
Returns
-------
vander3d : ndarray
The shape of the returned matrix is ``x.shape + (order,)``, where
:math:`order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1)`. The dtype will
be the same as the converted `x`, `y`, and `z`.
See Also
--------
polyvander, polyvander3d, polyval2d, polyval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((polyvander, polyvander, polyvander), (x, y, z), deg) | Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees `deg` and sample points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, then The pseudo-Vandermonde matrix is defined by .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = x^i * y^j * z^k, where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading indices of `V` index the points `(x, y, z)` and the last index encodes the powers of `x`, `y`, and `z`. If ``V = polyvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns of `V` correspond to the elements of a 3-D coefficient array `c` of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... and ``np.dot(V, c.flat)`` and ``polyval3d(x, y, z, c)`` will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D polynomials of the same degrees and sample points. Parameters ---------- x, y, z : array_like Arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. deg : list of ints List of maximum degrees of the form [x_deg, y_deg, z_deg]. Returns ------- vander3d : ndarray The shape of the returned matrix is ``x.shape + (order,)``, where :math:`order = (deg[0]+1)*(deg([1]+1)*(deg[2]+1)`. The dtype will be the same as the converted `x`, `y`, and `z`. See Also -------- polyvander, polyvander3d, polyval2d, polyval3d Notes ----- .. versionadded:: 1.7.0 |
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