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

Chebyshev(coef[, domain, window, symbol])

A Chebyshev series class.

Constants

chebdomain	An array object represents a multidimensional, homogeneous array of fixed-size items.
chebzero	An array object represents a multidimensional, homogeneous array of fixed-size items.
chebone	An array object represents a multidimensional, homogeneous array of fixed-size items.
chebx	An array object represents a multidimensional, homogeneous array of fixed-size items.

Arithmetic

chebadd(c1, c2)	Add one Chebyshev series to another.
chebsub(c1, c2)	Subtract one Chebyshev series from another.
chebmulx(c)	Multiply a Chebyshev series by x.
chebmul(c1, c2)	Multiply one Chebyshev series by another.
chebdiv(c1, c2)	Divide one Chebyshev series by another.
chebpow(c, pow[, maxpower])	Raise a Chebyshev series to a power.
chebval(x, c[, tensor])	Evaluate a Chebyshev series at points x.
chebval2d(x, y, c)	Evaluate a 2-D Chebyshev series at points (x, y).
chebval3d(x, y, z, c)	Evaluate a 3-D Chebyshev series at points (x, y, z).
chebgrid2d(x, y, c)	Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.
chebgrid3d(x, y, z, c)	Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z.

Calculus

chebder(c[, m, scl, axis])	Differentiate a Chebyshev series.
chebint(c[, m, k, lbnd, scl, axis])	Integrate a Chebyshev series.

Misc Functions

chebfromroots(roots)	Generate a Chebyshev series with given roots.
chebroots(c)	Compute the roots of a Chebyshev series.
chebvander(x, deg)	Pseudo-Vandermonde matrix of given degree.
chebvander2d(x, y, deg)	Pseudo-Vandermonde matrix of given degrees.
chebvander3d(x, y, z, deg)	Pseudo-Vandermonde matrix of given degrees.
chebgauss(deg)	Gauss-Chebyshev quadrature.
chebweight(x)	The weight function of the Chebyshev polynomials.
chebcompanion(c)	Return the scaled companion matrix of c.
chebfit(x, y, deg[, rcond, full, w])	Least squares fit of Chebyshev series to data.
chebpts1(npts)	Chebyshev points of the first kind.
chebpts2(npts)	Chebyshev points of the second kind.
chebtrim(c[, tol])	Remove "small" "trailing" coefficients from a polynomial.
chebline(off, scl)	Chebyshev series whose graph is a straight line.
cheb2poly(c)	Convert a Chebyshev series to a polynomial.
poly2cheb(pol)	Convert a polynomial to a Chebyshev series.
chebinterpolate(func, deg[, args])	Interpolate a function at the Chebyshev points of the first kind.

See also

numpy.polynomial

Notes

The implementations of multiplication, division, integration, and differentiation use the algebraic identities [1]:

\[\begin{split}T_n(x) = \frac{z^n + z^{-n}}{2} \\ z\frac{dx}{dz} = \frac{z - z^{-1}}{2}.\end{split}\]

where

\[x = \frac{z + z^{-1}}{2}.\]

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

[1]

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)