gen_hermite_rule

gen_hermite_rule, a Fortran90 code which generates a specific generalized Gauss-Hermite quadrature rule, based on user input.

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

The generalized Gauss Hermite quadrature rule is used as follows:

        Integral ( -oo < x < +oo ) |x-a|^alpha * exp( - b * ( x - a)^2 ) f(x) dx
      

        Sum ( 1 <= i <= order ) w(i) * f(x(i))
      

Usage:

gen_hermite_rule order alpha a b filename

• order is the number of points in the quadrature rule.
• alpha is the parameter for the generalized Gauss-Hermite quadrature rule. The value of alpha may be any real value greater than -1.0. Specifying alpha=0.0 results in the basic (non-generalized) rule.
• a is the center point (default 0);
• b is the scale factor (default 1);
• filename specifies the names of the output files: filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

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

Related Data and Programs:

gen_hermite_rule_test

f90_rule, a Fortran90 code which computes a quadrature rule which estimates the integral of a function f(x), which might be defined over a one dimensional region (a line) or more complex shapes such as a circle, a triangle, a quadrilateral, a polygon, or a higher dimensional region, and which might include an associated weight function w(x).

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:

• gen_hermite_rule.f90, the source code.
• gen_hermite_rule.sh, compiles the source code.