gegenbauer_rule


gegenbauer_rule, a FORTRAN90 code which generates a specific Gauss-Gegenbauer quadrature rule, based on user input.

The rule is written to three files for easy use as input to other programs.

The Gauss-Gegenbauer quadrature rule is used as follows:

        Integral ( A <= x <= B ) ((x-A)(B-X))^alpha f(x) dx
      
is to be approximated by
        Sum ( 1 <= i <= order ) w(i) * f(x(i))
      
where alpha is a real parameter greater than -1.

Usage:

gegenbauer_rule order alpha a b filename
where

Licensing:

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

Languages:

gegenbauer_rule is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

ALPERT_RULE, a FORTRAN90 code which can set up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

CCN_RULE, a FORTRAN90 code which defines a nested Clenshaw Curtis quadrature rule.

CHEBYSHEV1_RULE, a FORTRAN90 code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a FORTRAN90 code which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a FORTRAN90 code which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_CC, a FORTRAN90 code which estimates the Gegenbauer weighted integral of a function f(x) using a Clenshaw-Curtis approach.

GEGENBAUER_EXACTNESS, a FORTRAN90 code which checks the polynomial exactness of a Gauss-Gegenbauer rule.

gegenbauer_rule_test

GEN_HERMITE_RULE, a FORTRAN90 code which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a FORTRAN90 code which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a FORTRAN90 code which can compute and print a Gauss-Hermite quadrature rule.

JACOBI_RULE, a FORTRAN90 code which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a FORTRAN90 code which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a FORTRAN90 code which computes a Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, a FORTRAN90 code which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

LINE_NCO_RULE, a FORTRAN90 code which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

LOGNORMAL_RULE, a FORTRAN90 code which can compute and print a quadrature rule for functions of a variable whose logarithm is normally distributed.

PATTERSON_RULE, a FORTRAN90 code which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

PATTERSON_RULE_COMPUTE, a FORTRAN90 code which computes the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.

QUADRATURE_RULES_GEGENBAUER, a dataset directory which contains triples of files defining Gauss-Gegenbauer quadrature rules.

QUADRULE, a FORTRAN90 code which contains 1-dimensional quadrature rules.

TRUNCATED_NORMAL_RULE, a FORTRAN90 code which computes a quadrature rule for a normal distribution that has been truncated to [A,+oo), (-oo,B] or [A,B].

Reference:

  1. Milton Abramowitz, Irene Stegun,
    Handbook of Mathematical Functions,
    National Bureau of Standards, 1964,
    ISBN: 0-486-61272-4,
    LC: QA47.A34.
  2. Philip Davis, Philip Rabinowitz,
    Methods of Numerical Integration,
    Second Edition,
    Dover, 2007,
    ISBN: 0486453391,
    LC: QA299.3.D28.
  3. Sylvan Elhay, Jaroslav Kautsky,
    Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature,
    ACM Transactions on Mathematical Software,
    Volume 13, Number 4, December 1987, pages 399-415.
  4. Jaroslav Kautsky, Sylvan Elhay,
    Calculation of the Weights of Interpolatory Quadratures,
    Numerische Mathematik,
    Volume 40, 1982, pages 407-422.
  5. Roger Martin, James Wilkinson,
    The Implicit QL Algorithm,
    Numerische Mathematik,
    Volume 12, Number 5, December 1968, pages 377-383.
  6. Arthur Stroud, Don Secrest,
    Gaussian Quadrature Formulas,
    Prentice Hall, 1966,
    LC: QA299.4G3S7.

Source Code:


Last revised on 11 July 2020.