CHEBYSHEV_POLYNOMIAL
Chebyshev Polynomials

CHEBYSHEV_POLYNOMIAL is 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.

The Chebyshev polynomial T(n,x), or Chebyshev polynomial of the first kind, may be defined, for 0 <= n, and -1 <= x <= +1 by: 

        cos ( t ) = x
        T(n,x) = cos ( n * t )
For any value of x, T(n,x) may be evaluated by a three term recurrence: 
        T(0,x) = 1
        T(1,x) = x
        T(n+1,x) = 2x T(n,x) - T(n-1,x)

The Chebyshev polynomial U(n,x), or Chebyshev polynomial of the second kind, may be defined, for 0 <= n, and -1 <= x <= +1 by: 

        cos ( t ) = x
        U(n,x) = sin ( ( n + 1 ) t ) / sin ( t )
For any value of x, U(n,x) may be evaluated by a three term recurrence: 
        U(0,x) = 1
        U(1,x) = 2x
        U(n+1,x) = 2x U(n,x) - U(n-1,x)

The Chebyshev polynomial V(n,x), or Chebyshev polynomial of the third kind, may be defined, for 0 <= n, and -1 <= x <= +1 by: 

        cos ( t ) = x
        V(n,x) = cos ( (2n+1)*t/2) / cos ( t/2)
For any value of x, V(n,x) may be evaluated by a three term recurrence: 
        V(0,x) = 1
        V(1,x) = 2x-1
        V(n+1,x) = 2x V(n,x) - V(n-1,x)

The Chebyshev polynomial W(n,x), or Chebyshev polynomial of the fourth kind, may be defined, for 0 <= n, and -1 <= x <= +1 by: 

        cos ( t ) = x
        W(n,x) = sin((2*n+1)*t/2)/sin(t/2)
For any value of x, W(n,x) may be evaluated by a three term recurrence: 
        W(0,x) = 1
        W(1,x) = 2x+1
        W(n+1,x) = 2x W(n,x) - W(n-1,x)

Licensing:

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

Languages:

CHEBYSHEV_POLYNOMIAL is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

BERNSTEIN_POLYNOMIAL, a C++ library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

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

CHEBYSHEV_SERIES, 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.

CHEBYSHEV1_RULE, a C++ program which computes and prints a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a C++ program which compute and print a Gauss-Chebyshev type 2 quadrature rule.

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

GEGENBAUER_POLYNOMIAL, a C++ library which evaluates the Gegenbauer polynomial and associated functions.

HERMITE_POLYNOMIAL, a C++ library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

INT_EXACTNESS_CHEBYSHEV1, a C++ program which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.

INT_EXACTNESS_CHEBYSHEV2, a C++ program which tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.

JACOBI_POLYNOMIAL, a C++ library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a C++ library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a C++ library which evaluates the Legendre polynomial and associated functions.

LEGENDRE_SHIFTED_POLYNOMIAL, a C++ library which evaluates the shifted Legendre polynomial, with domain [0,1].

JACOBI_POLYNOMIAL, a C++ library which evaluates the Jacobi polynomial and associated functions.

LAGUERRE_POLYNOMIAL, a C++ library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LEGENDRE_POLYNOMIAL, a C++ library which evaluates the Legendre polynomial and associated functions.

LEGENDRE_SHIFTED_POLYNOMIAL, a C++ library which evaluates the shifted Legendre polynomial, with domain [0,1].

LOBATTO_POLYNOMIAL, a C++ library which evaluates Lobatto polynomials, similar to Legendre polynomials except that they are zero at both endpoints.

POLPAK, a C++ library which evaluates a variety of mathematical functions.

TEST_VALUES, a C++ library which supplies test values of various mathematical functions.

Reference:

  1. Theodore Chihara,
    An Introduction to Orthogonal Polynomials,
    Gordon and Breach, 1978,
    ISBN: 0677041500,
    LC: QA404.5 C44.
  2. Walter Gautschi,
    Orthogonal Polynomials: Computation and Approximation,
    Oxford, 2004,
    ISBN: 0-19-850672-4,
    LC: QA404.5 G3555.
  3. John Mason, David Handscomb,
    Chebyshev Polynomials,
    CRC Press, 2002,
    ISBN: 0-8493-035509,
    LC: QA404.5.M37.
  4. Frank Olver, Daniel Lozier, Ronald Boisvert, Charles Clark,
    NIST Handbook of Mathematical Functions,
    Cambridge University Press, 2010,
    ISBN: 978-0521192255,
    LC: QA331.N57.
  5. Gabor Szego,
    Orthogonal Polynomials,
    American Mathematical Society, 1992,
    ISBN: 0821810235,
    LC: QA3.A5.v23.

Source Code:

Examples and Tests:

List of Routines:

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

Last revised on 21 July 2015.