HERMITE_RULE_SS
Gauss-Hermite Quadrature Rules

HERMITE_RULE_SS is a C++ program which generates a specific Gauss-Hermite quadrature rule, based on user input, using a Stroud and Secrest algorithm.

The rule can be output as text in a standard programming language, or the data can be written to three files for easy use as input to other programs.

The Gauss-Hermite quadrature rule is designed to approximate integrals on infinite intervals.

The Gauss Hermite quadrature assumes that the integrand we are considering has a form like:

        Integral ( -oo < x < +oo ) w(x) * f(x) dx

where the factor w(x) is regarded as a weight factor.

We consider three variations of the rule, depending on the form of the weight factor w(x):

option = 0, the unweighted rule:

            Integral ( -oo < x < +oo ) f(x) dx

option = 1, the physicist weighted rule:

            Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx

option = 2, the probabilist weighted rule:

            Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx

The corresponding Gauss-Hermite rule that uses order points will approximate the integral by

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

where, confusingly, w(i) is a vector of quadrature weights, which has no connection with the w(x) weight function.

Usage:

hermite_rule_ss order option output

where

order is the number of points in the quadrature rule. A typical value might be 4, 8, or 16.

option specified the rule type:
a 0 value requests a rule for the unweighted integral:

            Integral ( -oo < x < +oo )         f(x) dx

a 1 value requests a rule for the physicist weighted integral:

            Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx

a 2 value requests a rule for the probabilist weighted integral:

            Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx

output specifies how the rule is to be reported:
- C++, print as C++ text;
- F77, print as FORTRAN77 text;
- F90, print as FORTRAN90 text;
- MAT, print as MATLAB text;
- file, written to three files, file_w.txt, file_x.txt, and file_r.txt, containing the weights, abscissas, and interval limits.

Licensing:

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

Languages:

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

Related Data and Programs:

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

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

CLENSHAW_CURTIS_RULE is a C++ program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, is a C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.

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

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

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

INT_EXACTNESS, is a C++ program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

INT_EXACTNESS_HERMITE, is a C++ program which checks the polynomial exactness of a Gauss-Hermite quadrature rule.

INTLIB is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

JACOBI_RULE, is a C++ program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, is a C++ program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, is a C++ program which computes a Gauss-Legendre quadrature rule.

PATTERSON_RULE, is a C++ program which computes a Gauss-Patterson quadrature rule.

PRODUCT_FACTOR is an C++ program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is a C++ program which constructs a product rule from identical 1D factor rules.

QUADPACK is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

QUADRATURE_RULES is a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_HERMITE is a dataset directory of triples of files defining standard Hermite quadrature rules.

QUADRULE is a C++ library which contains 1-dimensional quadrature rules.

TANH_SINH_RULE, a C++ program which computes and writes out a tanh-sinh quadrature rule of given order.

TEST_INT is a FORTRAN90 library which defines functions that may be used as test integrands for quadrature rules in 1D.

TEST_INT_HERMITE is a C++ library which defines test integrands for integration over (-oo,+oo).

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.
Arthur Stroud, Don Secrest,
Gaussian Quadrature Formulas,
Prentice Hall, 1966,
LC: QA299.4G3S7.

Source Code:

hermite_rule_ss.cpp, the source code.

Examples and Tests:

output_cpp_physicist.txt, printed output from the command


            hermite_rule_ss 4 1 C++

output_f77_physicist.txt, printed output from the command


            hermite_rule_ss 4 1 F77

output_f90_unweighted.txt, printed output from the command


            hermite_rule_ss 4 0 F90

output_f90_physicist.txt, printed output from the command


            hermite_rule_ss 4 1 F90

output_f90_probabilist.txt, printed output from the command


            hermite_rule_ss 4 2 F90

output_mat_physicist.txt, printed output from the command


            hermite_rule_ss 8 1 MAT

herm_o4_unweighted_r.txt, the region file created by the command


            hermite_rule_ss 4 0 herm_o4

herm_o4_unweighted_w.txt, the weight file created by the command


            hermite_rule_ss 4 0 herm_o4

herm_o4_unweighted_x.txt, the abscissa file created by the command
```
            hermite_rule_ss 4 0 herm_o4
          
```

herm_o4_physicist_r.txt, the region file created by the command


            hermite_rule_ss 4 1 herm_o4

herm_o4_physicist_w.txt, the weight file created by the command


            hermite_rule_ss 4 1 herm_o4

herm_o4_physicist_x.txt, the abscissa file created by the command


            hermite_rule_ss 4 1 herm_o4

herm_o4_probabilist_r.txt, the region file created by the command


            hermite_rule_ss 4 2 herm_o4

herm_o4_probabilist_w.txt, the weight file created by the command


            hermite_rule_ss 4 2 herm_o4

herm_o4_probabilist_x.txt, the abscissa file created by the command
```
            hermite_rule_ss 4 2 herm_o4
          
```

List of Routines:

MAIN is the main program for HERMITE_RULE.
DTABLE_WRITE0 writes information to a DTABLE file.
HERMITE_COMPUTE computes a Gauss-Hermite quadrature rule.
HERMITE_HANDLE computes the requested Gauss-Hermite rule and outputs it.
HERMITE_RECUR finds the value and derivative of a Hermite polynomial.
HERMITE_ROOT improves an approximate root of a Hermite polynomial.
R8_ABS returns the absolute value of an R8.
R8_EPSILON returns the R8 roundoff unit.
R8_GAMMA evaluates Gamma(X) for a real argument.
R8_HUGE returns a "huge" R8.
TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.

Last revised on 28 June 2009.

HERMITE_RULE_SS Gauss-Hermite Quadrature Rules