GEGENBAUER_POLYNOMIAL
Gegenbauer Polynomials

GEGENBAUER_POLYNOMIAL, a MATLAB library which evaluates Gegenbauer polynomials and associated functions.

The Gegenbauer polynomial C(n,alpha,x) can be defined by:

        C(0,alpha,x) = 1
        C(1,alpha,x) = 2 * alpha * x
        C(n,alpha,x) = (1/n) * ( 2*x*(n+alpha-1) * C(n-1,alpha,x) - (n+2*alpha-2) * C(n-2,alpha,x) )
      

The N zeroes of C(n,alpha,x) are the abscissas used for Gauss-Gegenbauer quadrature of the integral of a function F(X) with weight function (1-x^2)^(alpha-1/2) over the interval [-1,1].

The Gegenbauer polynomials are orthogonal under the inner product defined as weighted integration from -1 to 1:

        Integral ( -1 <= x <= 1 ) (1-x^2)^(alpha-1/2) * C(i,alpha,x) * C(j,alpha,x) dx 
          = 0 if i =/= j
          = pi * 2^(1-2*alpha) * Gamma(n+2*alpha) / n! / (n+alpha) / (Gamma(alpha))^2 if i = j.
      

Licensing:

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

Languages:

GEGENBAUER_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 MATLAB library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

CHEBYSHEV_POLYNOMIAL, a MATLAB 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.

GEGENBAUER_CC, a MATLAB library which estimates the Gegenbauer weighted integral of a function f(x) using a Clenshaw-Curtis approach.

GEGENBAUER_EXACTNESS, a MATLAB program which tests the monomial exactness of Gauss-Gegenbauer quadrature rules.

gegenbauer_polynomial_test

GEGENBAUER_RULE, a MATLAB program which can compute and print a Gauss-Gegenbauer quadrature rule.

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

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

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

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

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

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Source Code:

• gegenbauer_alpha_check.m, checks that the Gegenbauer parameter ALPHA is legal.
• gegenbauer_ek_compute.m, computes a Gegenbauer quadrature rule using the Elhay-Kautsky algorithm.
• gegenbauer_integral.m, evaluates the integral of a monomial with Gegenbauer weight.
• gegenbauer_polynomial_value.m, evaluates the Gegenbauer polynomials C(n,alpha,x).
• gegenbauer_polynomial_values.m, returns selected values of the Gegenbauer polynomials C(n,alpha,x).
• gegenbauer_ss_compute.m, computes a Gegenbauer quadrature rule using the Stroud-Secrest algorithm.
• gegenbauer_ss_recur.m finds the value and derivative of a Gegenbauer polynomial.
• gegenbauer_ss_root.m improves an approximate root of a Gegenbauer polynomial.
• hyper_2f1_values.m, returns selected values of the hypergeometric function F(A,B,C,X).
• imtqlx.m diagonalizes a symmetric tridiagonal matrix.
• r8_hyper_2f1.m, evaluates the hypergeometric function F(A,B,C,X).
• r8_uniform_ab.m, returns a scaled pseudorandom R8;
• r8vec_print.m, prints an R8VEC;
• timestamp.m, prints the current YMDHMS date as a time stamp.