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numpy.polynomial.chebyshev.Chebyshev

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¶

 Chebyshev(coef[, domain, window]) A Chebyshev series class.

## Arithmetic¶

 chebadd(c1, c2) Add one Chebyshev series to another. chebsub(c1, c2) Subtract one Chebyshev series from another. 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. 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. The weight function of the Chebyshev polynomials. 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. 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.

numpy.polynomial

## 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¶

1

A. T. Benjamin, et al., “Combinatorial Trigonometry with Chebyshev Polynomials,” Journal of Statistical Planning and Inference 14, 2008 (preprint: https://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4)