HERMITE_RULE
Gauss-Hermite Quadrature Rules

HERMITE_RULE is a FORTRAN90 program 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:

scale=0, then C = 1; this is the standard choice for Gauss-Hermite quadrature.
scale=1, then C is a normalization factor so that f(x)=1 will integrate to 1. This implies in turn that the weights will sum to 1. This choice is appropriate when using the rule to compute probabilities.

Usage:

hermite_rule n a b scale filename

where

n is the number of points in the quadrature rule.
a is the center point (default 0);
b is the scale factor (default 1);
scale is the normalization option (0/1). If 1, then the weights are normalized to have unit sum;
filename specifies the output filenames: filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL 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:

ALPERT_RULE, a FORTRAN90 library 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 program which defines a nested Clenshaw Curtis quadrature rule.

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

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

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

GEGENBAUER_RULE, a FORTRAN90 program which can compute and print a Gauss-Gegenbauer quadrature rule.

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

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

HERMITE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

HERMITE_POLYNOMIAL, a FORTRAN90 library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

HERMITE_TEST_INT, a FORTRAN90 library which defines test integrands for Hermite integrals with interval (-oo,+oo) and density exp(-x^2).

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

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

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

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

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

PATTERSON_RULE, a FORTRAN90 program 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 program 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_HERMITE_PHYSICIST, 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 FORTRAN90 library which contains 1-dimensional quadrature rules.

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

Reference:

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

Source Code:

hermite_rule.f90, the source code.

Examples and Tests:

HERM_O4 is a Hermite rule of order 4, created by the command

        hermite_rule 4 0.0 1.0 0 herm_o4

herm_o4_r.txt, the region file;
herm_o4_w.txt, the weight file;
herm_o4_x.txt, the abscissa file;

List of Routines:

MAIN is the main program for HERMITE_RULE.
CDGQF computes a Gauss quadrature formula with default A, B and simple knots.
CGQF computes knots and weights of a Gauss quadrature formula.
CH_CAP capitalizes a single character.
CH_EQI is a case insensitive comparison of two characters for equality.
CH_TO_DIGIT returns the integer value of a base 10 digit.
CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
GET_UNIT returns a free FORTRAN unit number.
IMTQLX diagonalizes a symmetric tridiagonal matrix.
PARCHK checks parameters ALPHA and BETA for classical weight functions.
R8_GAMMA evaluates Gamma(X) for a real argument.
R8_HUGE returns a very large R8.
R8MAT_WRITE writes an R8MAT file.
RULE_WRITE writes a quadrature rule to a file.
S_TO_I4 reads an I4 from a string.
S_TO_R8 reads an R8 from a string.
SCQF scales a quadrature formula to a nonstandard interval.
SGQF computes knots and weights of a Gauss Quadrature formula.
TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.

Last revised on 28 December 2010.

HERMITE_RULE Gauss-Hermite Quadrature Rules