GEGENBAUER_POLYNOMIAL
Gegenbauer Polynomials

GEGENBAUER_POLYNOMIAL is a Python 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 Python library which evaluates the Bernstein polynomials, useful for uniform approximation of functions;

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

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

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POLPAK, a Python library which evaluates a variety of mathematical functions.

Source Code:

• gamma_values.py, returns selected values of the Gamma function.
• gegenbauer_alpha_check.py, checks that the Gegenbauer parameter ALPHA is legal.
• gegenbauer_ek_compute.py, computes a Gegenbauer quadrature rule using the Elhay-Kautsky algorithm.
• gegenbauer_integral.py, evaluates the integral of a monomial with Gegenbauer weight.
• gegenbauer_polynomial_value.py, evaluates the Gegenbauer polynomials C(n,alpha,x).
• gegenbauer_polynomial_values.py, returns selected values of the Gegenbauer polynomials C(n,alpha,x).
• gegenbauer_ss_compute.py, computes a Gegenbauer quadrature rule using the Stroud-Secrest algorithm.
• hyper_2f1_values.py, returns selected values of the hypergeometric function F(A,B,C,X).
• imtqlx.py diagonalizes a symmetric tridiagonal matrix.
• psi_values.py returns some values of the Psi or Digamma function.
• r8_epsilon.py, returns the R8 roundoff unit.
• r8_gamma.py, returns the gamma function of an R8.
• r8_hyper_2f1.py, evaluates the hypergeometric function F(A,B,C,X).
• r8_psi.py, returns the Psi function;
• r8_uniform_ab.py, returns a scaled pseudorandom R8;
• r8vec_print.py, prints an R8VEC.
• r8vec_reverse.py, copies the elements of an R8VEC in reverse order.
• timestamp.py, prints the current YMDHMS date as a time stamp.

Examples and Tests:

• gegenbauer_polynomial_test.py, calls all the tests.
• gegenbauer_polynomial_test.sh, runs all the tests.
• gegenbauer_polynomial_test.txt, the output file.

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