# chebyshev_series

chebyshev_series, a MATLAB code which can evaluate a Chebyshev series approximating a function f(x), while efficiently computing one, two or three derivatives of the series, which approximate f'(x), f''(x), and f'''(x), by Manfred Zimmer.

Note that this library does not compute a Chebyshev series; it assumes that the series has already been computed, and offers an efficient means of evaluating the series and its derivatives simultaneously.

### Languages:

chebyshev_series is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

chebyshev, a MATLAB code which computes the Chebyshev interpolant/approximant to a given function over an interval.

chebyshev_interp_1d, a MATLAB code which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

chebyshev_polynomial, a MATLAB code which considers the Chebyshev polynomials T(i,x), U(i,x), V(i,x) and W(i,x). Functions are provided to evaluate the polynomials, determine their zeros, produce their polynomial coefficients, produce related quadrature rules, project other functions onto these polynomial bases, and integrate double and triple products of the polynomials.

clausen, a MATLAB code which evaluates a Chebyshev approximant to the Clausen function Cl2(x).

fn, a MATLAB code which approximates elementary and special functions using Chebyshev polynomials; functions include Airy, Bessel I, J, K and Y, beta, confluent hypergeometric, error, gamma, log gamma, Pochhammer, Spence; integrals include hyperbolic cosine, cosine, Dawson, exponential, logarithmic, hyperbolic sine, sine; by Wayne Fullerton.

polpak, a MATLAB code which evaluates a variety of mathematical functions, including Chebyshev, Gegenbauer, Hermite, Jacobi, Laguerre, Legendre polynomials, and the Collatz sequence.

toms446, a MATLAB code which manipulates Chebyshev series for interpolation and approximation; this is ACM TOMS algorithm 446, by Roger Broucke.

Manfred Zimmer

### Reference:

1. Charles Clenshaw,
Mathematical Tables, Volume 5,
Chebyshev series for mathematical functions,
London, 1962.
2. Gerhard Maess,
Vorlesungen ueber Numerische Mathematik II, Analysis,
ISBN: 978-3764318840,
LC: QA297.M325.
3. Francis Smith,
An algorithm for summing orthogonal polynomial series and their derivatives with applications to curve-fitting and interpolation,
Mathematics of Computation,
Volume 19, Number 89, 1965, pages 33-36.

### Source Code:

• echebser0.m, evaluates a Chebyshev series.
• echebser1.m, evaluates a Chebyshev series and first derivative.
• echebser2.m, evaluates a Chebyshev series and two derivatives.
• echebser3.m, evaluates a Chebyshev series and three derivatives.
• echebser4.m, evaluates a Chebyshev series and four derivatives.
• evenchebser0.m, evaluates an even Chebyshev series.
• evenchebser1.m, evaluates an even Chebyshev series and first derivative.
• evenchebser2.m, evaluates an even Chebyshev series and first two derivatives.
• oddchebser0.m, evaluates an odd Chebyshev series.
• oddchebser1.m, evaluates an odd Chebyshev series and the first derivative.
• oddchebser2.m, evaluates an odd Chebyshev series and first two derivatives.

Last revised on 11 December 2018.