hermite_rule


hermite_rule, a MATLAB code which generates a specific Gauss-Hermite quadrature rule, based on user input.

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

The Gauss-Hermite quadrature rule is used as follows:

        c * Integral ( -oo < x < +oo ) f(x) exp ( - b * ( x - a )^2 ) dx
      
is to be approximated by
        Sum ( 1 <= i <= order ) w(i) * f(x(i))
      
Generally, a Gauss-Hermite quadrature rule of n points will produce the exact integral when f(x) is a polynomial of degree 2n-1 or less.

The value of C in front of the integral depends on the user's choice of the SCALE parameter:

Usage:

hermite_rule ( order, a, b, scale 'filename' )
where

Licensing:

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

Languages:

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

Related Data and Programs:

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ccn_rule, a MATLAB code which defines a nested Clenshaw Curtis quadrature rule.

chebyshev1_rule, a MATLAB code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

chebyshev2_rule, a MATLAB code which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

clenshaw_curtis_rule, a MATLAB code which defines a Clenshaw Curtis quadrature rule.

gegenbauer_rule, a MATLAB code which can compute and print a Gauss-Gegenbauer quadrature rule.

gen_hermite_rule, a MATLAB code which can compute and print a generalized Gauss-Hermite quadrature rule.

gen_laguerre_rule, a MATLAB code which can compute and print a generalized Gauss-Laguerre quadrature rule.

hermite_exactness, a MATLAB code which tests the polynomial exactness of Gauss-Hermite quadrature rules.

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

hermite_rule_test

jacobi_rule, a MATLAB code which can compute and print a Gauss-Jacobi quadrature rule.

laguerre_rule, a MATLAB code which can compute and print a Gauss-Laguerre quadrature rule.

legendre_rule, a MATLAB code which computes a Gauss-Legendre quadrature rule.

line_felippa_rule, a MATLAB code which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

patterson_rule, a MATLAB code which computes a Gauss-Patterson quadrature rule.

quadrature_rules_hermite_physicis, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2).

quadrature_rules_hermite_probabilist, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2/2).

quadrature_rules_hermite_unweighted, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function 1.

quadrule, a MATLAB code which contains 1-dimensional quadrature rules.

truncated_normal_rule, a MATLAB code which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian 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 29 January 2019.