CHEBYSHEV_SERIES
Evaluate Chebyshev Series with Derivatives

CHEBYSHEV_SERIES is a C++ library 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.

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

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 C++ library which computes the Chebyshev interpolant/approximant to a given function over an interval.

CHEBYSHEV_INTERP_1D, a C++ library which determines the combination of Chebyshev polynomials which interpolates a set of data, so that p(x(i)) = y(i).

CHEBYSHEV_POLYNOMIAL, a C++ library 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 C++ library which evaluates a Chebyshev interpolant to the Clausen function Cl2(x).

FN, a C++ library 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 C++ library which evaluates a variety of mathematical functions, including Chebyshev, Gegenbauer, Hermite, Jacobi, Laguerre, Legendre polynomials, and the Collatz sequence.

TOMS446, a C++ library which manipulates Chebyshev series for interpolation and approximation; this is ACM TOMS algorithm 446, by Roger Broucke.

Author:

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,
   Berlin, Akademie_Verlag, 1984-1988,
   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:

• chebyshev_series.cpp, the source code.
• chebyshev_series.hpp, the include file.

Examples and Tests:

• chebyshev_series_test.cpp, a sample calling program.
• chebyshev_series_test.txt, the output file.

List of Routines:

• ECHEBSER0 evaluates a Chebyshev series.
• ECHEBSER1 evaluates a Chebyshev series and first derivative.
• ECHEBSER2 evaluates a Chebyshev series and two derivatives.
• ECHEBSER3 evaluates a Chebyshev series and three derivatives.
• ECHEBSER4 evaluates a Chebyshev series and four derivatives.
• EVENCHEBSER0 evaluates an even Chebyshev series.
• EVENCHEBSER1 evaluates an even Chebyshev series and first derivative.
• EVENCHEBSER2 evaluates an even Chebyshev series and first two derivatives.
• ODDCHEBSER0 evaluates an odd Chebyshev series.
• ODDCHEBSER1 evaluates an odd Chebyshev series and the first derivative.
• ODDCHEBSER2 evaluates an odd Chebyshev series and first two derivatives.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.