New in version 1.4.0.
Chebyshev Series (numpy.polynomial.chebyshev
)#
This module provides a number of objects (mostly functions) useful for
dealing with Chebyshev series, including a Chebyshev
class that
encapsulates the usual arithmetic operations. (General information
on how this module represents and works with such polynomials is in the
docstring for its “parent” sub-package, numpy.polynomial
).
Classes#
|
A Chebyshev series class. |
Constants#
An array object represents a multidimensional, homogeneous array of fixed-size items. |
|
An array object represents a multidimensional, homogeneous array of fixed-size items. |
|
An array object represents a multidimensional, homogeneous array of fixed-size items. |
|
An array object represents a multidimensional, homogeneous array of fixed-size items. |
Arithmetic#
|
Add one Chebyshev series to another. |
|
Subtract one Chebyshev series from another. |
|
Multiply a Chebyshev series by x. |
|
Multiply one Chebyshev series by another. |
|
Divide one Chebyshev series by another. |
|
Raise a Chebyshev series to a power. |
|
Evaluate a Chebyshev series at points x. |
|
Evaluate a 2-D Chebyshev series at points (x, y). |
|
Evaluate a 3-D Chebyshev series at points (x, y, z). |
|
Evaluate a 2-D Chebyshev series on the Cartesian product of x and y. |
|
Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z. |
Calculus#
Misc Functions#
|
Generate a Chebyshev series with given roots. |
|
Compute the roots of a Chebyshev series. |
|
Pseudo-Vandermonde matrix of given degree. |
|
Pseudo-Vandermonde matrix of given degrees. |
|
Pseudo-Vandermonde matrix of given degrees. |
|
Gauss-Chebyshev quadrature. |
|
The weight function of the Chebyshev polynomials. |
Return the scaled companion matrix of c. |
|
|
Least squares fit of Chebyshev series to data. |
|
Chebyshev points of the first kind. |
|
Chebyshev points of the second kind. |
|
Remove "small" "trailing" coefficients from a polynomial. |
|
Chebyshev series whose graph is a straight line. |
|
Convert a Chebyshev series to a polynomial. |
|
Convert a polynomial to a Chebyshev series. |
|
Interpolate a function at the Chebyshev points of the first kind. |
See also#
Notes#
The implementations of multiplication, division, integration, and differentiation use the algebraic identities [1]:
where
These identities allow a Chebyshev series to be expressed as a finite, symmetric Laurent series. In this module, this sort of Laurent series is referred to as a “z-series.”
References#
A. T. Benjamin, et al., “Combinatorial Trigonometry with Chebyshev Polynomials,” Journal of Statistical Planning and Inference 14, 2008 (https://web.archive.org/web/20080221202153/https://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4)