2083 lines
61 KiB
Python
2083 lines
61 KiB
Python
"""
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====================================================
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Chebyshev Series (:mod:`numpy.polynomial.chebyshev`)
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====================================================
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This module provides a number of objects (mostly functions) useful for
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dealing with Chebyshev series, including a `Chebyshev` class that
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encapsulates the usual arithmetic operations. (General information
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on how this module represents and works with such polynomials is in the
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docstring for its "parent" sub-package, `numpy.polynomial`).
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Classes
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-------
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.. autosummary::
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:toctree: generated/
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Chebyshev
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Constants
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---------
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.. autosummary::
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:toctree: generated/
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chebdomain
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chebzero
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chebone
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chebx
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Arithmetic
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----------
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.. autosummary::
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:toctree: generated/
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chebadd
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chebsub
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chebmulx
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chebmul
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chebdiv
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chebpow
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chebval
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chebval2d
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chebval3d
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chebgrid2d
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chebgrid3d
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Calculus
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--------
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.. autosummary::
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:toctree: generated/
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chebder
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chebint
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Misc Functions
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--------------
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.. autosummary::
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:toctree: generated/
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chebfromroots
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chebroots
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chebvander
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chebvander2d
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chebvander3d
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chebgauss
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chebweight
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chebcompanion
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chebfit
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chebpts1
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chebpts2
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chebtrim
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chebline
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cheb2poly
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poly2cheb
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chebinterpolate
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See also
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--------
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`numpy.polynomial`
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Notes
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-----
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The implementations of multiplication, division, integration, and
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differentiation use the algebraic identities [1]_:
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.. math::
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T_n(x) = \\frac{z^n + z^{-n}}{2} \\\\
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z\\frac{dx}{dz} = \\frac{z - z^{-1}}{2}.
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where
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.. math:: x = \\frac{z + z^{-1}}{2}.
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These identities allow a Chebyshev series to be expressed as a finite,
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symmetric Laurent series. In this module, this sort of Laurent series
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is referred to as a "z-series."
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References
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----------
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.. [1] A. T. Benjamin, et al., "Combinatorial Trigonometry with Chebyshev
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Polynomials," *Journal of Statistical Planning and Inference 14*, 2008
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(https://web.archive.org/web/20080221202153/https://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4)
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"""
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import numpy as np
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import numpy.linalg as la
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from numpy.core.multiarray import normalize_axis_index
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from . import polyutils as pu
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from ._polybase import ABCPolyBase
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__all__ = [
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'chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline', 'chebadd',
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'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow', 'chebval',
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'chebder', 'chebint', 'cheb2poly', 'poly2cheb', 'chebfromroots',
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'chebvander', 'chebfit', 'chebtrim', 'chebroots', 'chebpts1',
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'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d', 'chebgrid2d',
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'chebgrid3d', 'chebvander2d', 'chebvander3d', 'chebcompanion',
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'chebgauss', 'chebweight', 'chebinterpolate']
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chebtrim = pu.trimcoef
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#
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# A collection of functions for manipulating z-series. These are private
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# functions and do minimal error checking.
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#
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def _cseries_to_zseries(c):
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"""Convert Chebyshev series to z-series.
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Convert a Chebyshev series to the equivalent z-series. The result is
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never an empty array. The dtype of the return is the same as that of
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the input. No checks are run on the arguments as this routine is for
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internal use.
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Parameters
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----------
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c : 1-D ndarray
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Chebyshev coefficients, ordered from low to high
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Returns
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-------
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zs : 1-D ndarray
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Odd length symmetric z-series, ordered from low to high.
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"""
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n = c.size
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zs = np.zeros(2*n-1, dtype=c.dtype)
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zs[n-1:] = c/2
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return zs + zs[::-1]
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def _zseries_to_cseries(zs):
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"""Convert z-series to a Chebyshev series.
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Convert a z series to the equivalent Chebyshev series. The result is
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never an empty array. The dtype of the return is the same as that of
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the input. No checks are run on the arguments as this routine is for
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internal use.
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Parameters
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----------
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zs : 1-D ndarray
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Odd length symmetric z-series, ordered from low to high.
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Returns
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-------
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c : 1-D ndarray
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Chebyshev coefficients, ordered from low to high.
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"""
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n = (zs.size + 1)//2
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c = zs[n-1:].copy()
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c[1:n] *= 2
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return c
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def _zseries_mul(z1, z2):
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"""Multiply two z-series.
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Multiply two z-series to produce a z-series.
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Parameters
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----------
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z1, z2 : 1-D ndarray
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The arrays must be 1-D but this is not checked.
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Returns
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-------
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product : 1-D ndarray
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The product z-series.
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Notes
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-----
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This is simply convolution. If symmetric/anti-symmetric z-series are
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denoted by S/A then the following rules apply:
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S*S, A*A -> S
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S*A, A*S -> A
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"""
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return np.convolve(z1, z2)
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def _zseries_div(z1, z2):
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"""Divide the first z-series by the second.
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Divide `z1` by `z2` and return the quotient and remainder as z-series.
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Warning: this implementation only applies when both z1 and z2 have the
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same symmetry, which is sufficient for present purposes.
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Parameters
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----------
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z1, z2 : 1-D ndarray
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The arrays must be 1-D and have the same symmetry, but this is not
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checked.
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Returns
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-------
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(quotient, remainder) : 1-D ndarrays
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Quotient and remainder as z-series.
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Notes
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-----
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This is not the same as polynomial division on account of the desired form
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of the remainder. If symmetric/anti-symmetric z-series are denoted by S/A
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then the following rules apply:
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S/S -> S,S
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A/A -> S,A
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The restriction to types of the same symmetry could be fixed but seems like
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unneeded generality. There is no natural form for the remainder in the case
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where there is no symmetry.
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"""
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z1 = z1.copy()
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z2 = z2.copy()
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lc1 = len(z1)
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lc2 = len(z2)
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if lc2 == 1:
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z1 /= z2
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return z1, z1[:1]*0
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elif lc1 < lc2:
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return z1[:1]*0, z1
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else:
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dlen = lc1 - lc2
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scl = z2[0]
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z2 /= scl
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quo = np.empty(dlen + 1, dtype=z1.dtype)
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i = 0
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j = dlen
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while i < j:
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r = z1[i]
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quo[i] = z1[i]
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quo[dlen - i] = r
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tmp = r*z2
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z1[i:i+lc2] -= tmp
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z1[j:j+lc2] -= tmp
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i += 1
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j -= 1
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r = z1[i]
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quo[i] = r
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tmp = r*z2
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z1[i:i+lc2] -= tmp
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quo /= scl
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rem = z1[i+1:i-1+lc2].copy()
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return quo, rem
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def _zseries_der(zs):
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"""Differentiate a z-series.
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The derivative is with respect to x, not z. This is achieved using the
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chain rule and the value of dx/dz given in the module notes.
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Parameters
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----------
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zs : z-series
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The z-series to differentiate.
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Returns
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-------
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derivative : z-series
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The derivative
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Notes
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-----
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The zseries for x (ns) has been multiplied by two in order to avoid
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using floats that are incompatible with Decimal and likely other
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specialized scalar types. This scaling has been compensated by
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multiplying the value of zs by two also so that the two cancels in the
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division.
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"""
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n = len(zs)//2
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ns = np.array([-1, 0, 1], dtype=zs.dtype)
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zs *= np.arange(-n, n+1)*2
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d, r = _zseries_div(zs, ns)
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return d
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def _zseries_int(zs):
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"""Integrate a z-series.
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The integral is with respect to x, not z. This is achieved by a change
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of variable using dx/dz given in the module notes.
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Parameters
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----------
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zs : z-series
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The z-series to integrate
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Returns
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-------
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integral : z-series
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The indefinite integral
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Notes
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-----
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The zseries for x (ns) has been multiplied by two in order to avoid
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using floats that are incompatible with Decimal and likely other
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specialized scalar types. This scaling has been compensated by
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dividing the resulting zs by two.
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"""
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n = 1 + len(zs)//2
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ns = np.array([-1, 0, 1], dtype=zs.dtype)
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zs = _zseries_mul(zs, ns)
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div = np.arange(-n, n+1)*2
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zs[:n] /= div[:n]
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zs[n+1:] /= div[n+1:]
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zs[n] = 0
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return zs
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#
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# Chebyshev series functions
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#
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def poly2cheb(pol):
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"""
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Convert a polynomial to a Chebyshev series.
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Convert an array representing the coefficients of a polynomial (relative
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to the "standard" basis) ordered from lowest degree to highest, to an
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array of the coefficients of the equivalent Chebyshev series, ordered
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from lowest to highest degree.
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Parameters
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----------
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pol : array_like
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1-D array containing the polynomial coefficients
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Returns
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-------
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c : ndarray
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1-D array containing the coefficients of the equivalent Chebyshev
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series.
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See Also
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--------
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cheb2poly
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Notes
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-----
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The easy way to do conversions between polynomial basis sets
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is to use the convert method of a class instance.
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Examples
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--------
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>>> from numpy import polynomial as P
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>>> p = P.Polynomial(range(4))
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>>> p
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Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
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>>> c = p.convert(kind=P.Chebyshev)
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>>> c
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Chebyshev([1. , 3.25, 1. , 0.75], domain=[-1., 1.], window=[-1., 1.])
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>>> P.chebyshev.poly2cheb(range(4))
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array([1. , 3.25, 1. , 0.75])
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"""
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[pol] = pu.as_series([pol])
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deg = len(pol) - 1
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res = 0
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for i in range(deg, -1, -1):
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res = chebadd(chebmulx(res), pol[i])
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return res
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def cheb2poly(c):
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"""
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Convert a Chebyshev series to a polynomial.
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Convert an array representing the coefficients of a Chebyshev series,
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ordered from lowest degree to highest, to an array of the coefficients
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of the equivalent polynomial (relative to the "standard" basis) ordered
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from lowest to highest degree.
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Parameters
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----------
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c : array_like
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1-D array containing the Chebyshev series coefficients, ordered
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from lowest order term to highest.
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Returns
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-------
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pol : ndarray
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1-D array containing the coefficients of the equivalent polynomial
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(relative to the "standard" basis) ordered from lowest order term
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to highest.
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See Also
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--------
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poly2cheb
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Notes
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-----
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The easy way to do conversions between polynomial basis sets
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is to use the convert method of a class instance.
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Examples
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--------
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>>> from numpy import polynomial as P
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>>> c = P.Chebyshev(range(4))
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>>> c
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Chebyshev([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
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>>> p = c.convert(kind=P.Polynomial)
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>>> p
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Polynomial([-2., -8., 4., 12.], domain=[-1., 1.], window=[-1., 1.])
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>>> P.chebyshev.cheb2poly(range(4))
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array([-2., -8., 4., 12.])
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"""
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from .polynomial import polyadd, polysub, polymulx
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[c] = pu.as_series([c])
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n = len(c)
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if n < 3:
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return c
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else:
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c0 = c[-2]
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c1 = c[-1]
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# i is the current degree of c1
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for i in range(n - 1, 1, -1):
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tmp = c0
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c0 = polysub(c[i - 2], c1)
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c1 = polyadd(tmp, polymulx(c1)*2)
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return polyadd(c0, polymulx(c1))
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#
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# These are constant arrays are of integer type so as to be compatible
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# with the widest range of other types, such as Decimal.
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#
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# Chebyshev default domain.
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chebdomain = np.array([-1, 1])
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# Chebyshev coefficients representing zero.
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chebzero = np.array([0])
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# Chebyshev coefficients representing one.
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chebone = np.array([1])
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# Chebyshev coefficients representing the identity x.
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chebx = np.array([0, 1])
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def chebline(off, scl):
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"""
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Chebyshev series whose graph is a straight line.
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Parameters
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----------
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off, scl : scalars
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The specified line is given by ``off + scl*x``.
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Returns
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-------
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y : ndarray
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This module's representation of the Chebyshev series for
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``off + scl*x``.
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See Also
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--------
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numpy.polynomial.polynomial.polyline
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numpy.polynomial.legendre.legline
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numpy.polynomial.laguerre.lagline
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numpy.polynomial.hermite.hermline
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numpy.polynomial.hermite_e.hermeline
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Examples
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--------
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>>> import numpy.polynomial.chebyshev as C
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>>> C.chebline(3,2)
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array([3, 2])
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>>> C.chebval(-3, C.chebline(3,2)) # should be -3
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-3.0
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"""
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if scl != 0:
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return np.array([off, scl])
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else:
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return np.array([off])
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def chebfromroots(roots):
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"""
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Generate a Chebyshev series with given roots.
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The function returns the coefficients of the polynomial
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.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
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in Chebyshev form, where the `r_n` are the roots specified in `roots`.
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If a zero has multiplicity n, then it must appear in `roots` n times.
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For instance, if 2 is a root of multiplicity three and 3 is a root of
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multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
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roots can appear in any order.
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If the returned coefficients are `c`, then
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.. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x)
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The coefficient of the last term is not generally 1 for monic
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polynomials in Chebyshev form.
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Parameters
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----------
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roots : array_like
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Sequence containing the roots.
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Returns
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-------
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out : ndarray
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1-D array of coefficients. If all roots are real then `out` is a
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real array, if some of the roots are complex, then `out` is complex
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even if all the coefficients in the result are real (see Examples
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below).
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See Also
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--------
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numpy.polynomial.polynomial.polyfromroots
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numpy.polynomial.legendre.legfromroots
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numpy.polynomial.laguerre.lagfromroots
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numpy.polynomial.hermite.hermfromroots
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numpy.polynomial.hermite_e.hermefromroots
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Examples
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--------
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>>> import numpy.polynomial.chebyshev as C
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>>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis
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array([ 0. , -0.25, 0. , 0.25])
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>>> j = complex(0,1)
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>>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis
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array([1.5+0.j, 0. +0.j, 0.5+0.j])
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"""
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return pu._fromroots(chebline, chebmul, roots)
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def chebadd(c1, c2):
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"""
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Add one Chebyshev series to another.
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Returns the sum of two Chebyshev series `c1` + `c2`. The arguments
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are sequences of coefficients ordered from lowest order term to
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highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
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Parameters
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----------
|
|
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 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)
|
|
|
|
|
|
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
|
|
|
|
|
|
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)
|
|
|
|
|
|
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
|
|
|
|
|
|
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)
|
|
|
|
|
|
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
|
|
|
|
|
|
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
|
|
|
|
|
|
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
|
|
|
|
|
|
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)
|
|
|
|
|
|
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)
|
|
|
|
|
|
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)
|
|
|
|
|
|
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)
|
|
|
|
|
|
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 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)
|
|
|
|
|
|
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)
|
|
|
|
|
|
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)
|
|
|
|
|
|
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
|
|
|
|
|
|
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
|
|
|
|
|
|
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
|
|
|
|
|
|
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
|
|
|
|
|
|
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
|
|
|
|
|
|
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)
|
|
|
|
|
|
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 series class
|
|
#
|
|
|
|
class Chebyshev(ABCPolyBase):
|
|
"""A Chebyshev series class.
|
|
|
|
The Chebyshev class provides the standard Python numerical methods
|
|
'+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
|
|
methods listed below.
|
|
|
|
Parameters
|
|
----------
|
|
coef : array_like
|
|
Chebyshev coefficients in order of increasing degree, i.e.,
|
|
``(1, 2, 3)`` gives ``1*T_0(x) + 2*T_1(x) + 3*T_2(x)``.
|
|
domain : (2,) array_like, optional
|
|
Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
|
|
to the interval ``[window[0], window[1]]`` by shifting and scaling.
|
|
The default value is [-1, 1].
|
|
window : (2,) array_like, optional
|
|
Window, see `domain` for its use. The default value is [-1, 1].
|
|
|
|
.. versionadded:: 1.6.0
|
|
symbol : str, optional
|
|
Symbol used to represent the independent variable in string
|
|
representations of the polynomial expression, e.g. for printing.
|
|
The symbol must be a valid Python identifier. Default value is 'x'.
|
|
|
|
.. versionadded:: 1.24
|
|
|
|
"""
|
|
# Virtual Functions
|
|
_add = staticmethod(chebadd)
|
|
_sub = staticmethod(chebsub)
|
|
_mul = staticmethod(chebmul)
|
|
_div = staticmethod(chebdiv)
|
|
_pow = staticmethod(chebpow)
|
|
_val = staticmethod(chebval)
|
|
_int = staticmethod(chebint)
|
|
_der = staticmethod(chebder)
|
|
_fit = staticmethod(chebfit)
|
|
_line = staticmethod(chebline)
|
|
_roots = staticmethod(chebroots)
|
|
_fromroots = staticmethod(chebfromroots)
|
|
|
|
@classmethod
|
|
def interpolate(cls, func, deg, domain=None, args=()):
|
|
"""Interpolate a function at the Chebyshev points of the first kind.
|
|
|
|
Returns the series that interpolates `func` at the Chebyshev points of
|
|
the first kind scaled and shifted to the `domain`. The resulting series
|
|
tends to a minmax approximation of `func` when the function is
|
|
continuous in the domain.
|
|
|
|
.. versionadded:: 1.14.0
|
|
|
|
Parameters
|
|
----------
|
|
func : function
|
|
The function to be interpolated. 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.
|
|
domain : {None, [beg, end]}, optional
|
|
Domain over which `func` is interpolated. The default is None, in
|
|
which case the domain is [-1, 1].
|
|
args : tuple, optional
|
|
Extra arguments to be used in the function call. Default is no
|
|
extra arguments.
|
|
|
|
Returns
|
|
-------
|
|
polynomial : Chebyshev instance
|
|
Interpolating Chebyshev instance.
|
|
|
|
Notes
|
|
-----
|
|
See `numpy.polynomial.chebfromfunction` for more details.
|
|
|
|
"""
|
|
if domain is None:
|
|
domain = cls.domain
|
|
xfunc = lambda x: func(pu.mapdomain(x, cls.window, domain), *args)
|
|
coef = chebinterpolate(xfunc, deg)
|
|
return cls(coef, domain=domain)
|
|
|
|
# Virtual properties
|
|
domain = np.array(chebdomain)
|
|
window = np.array(chebdomain)
|
|
basis_name = 'T'
|