gegenbauer_polynomial

gegenbauer_polynomial, a MATLAB code 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 information on this web page is distributed under the MIT license.

Languages:

gegenbauer_polynomial is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

Related Data and Programs:

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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.
• gegenbauer_to_monomial_matrix.m converts Gegenbauer polynomial coefficients to monomial basis.
• 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).
• r8vec_print.m, prints an R8VEC;