HERMITE_RULE
Gauss-Hermite Quadrature Rules

HERMITE_RULE is an executable FORTRAN90 program, using double precision arithmetic, which can generate a specific Gauss-Hermite quadrature rule, based on user input.

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 ) exp(-x*x) * f(x) dx

where the factor exp(-x*x) is regarded as a weight factor.

The standard Gauss Hermite quadrature rule is used as follows:

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

is to be approximated by

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

Although the standard rule is defined in terms of the product of the integral weight function exp(-x*x) and a function f(x), there may be cases when it is more convenient to think that we are simply approximating

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

The standard rule can easily be modified, by adjusting the weights, so that the computation can be done in this form. The program allows the user to specify, through the parameter OPTION, whether the standard rule is to be computed (OPTION=0), or the modified rule (OPTION=1).

The modified Gauss-Hermite quadrature rule is used as follows:

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

is to be approximated by

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

Usage:

hermite_rule order option output

order

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

option

a 0 value requests the "standard" integration rule for

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

a 1 value requests the "modified" integration rule for

            Integral ( -oo < x < +oo )         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.

Related Data and Programs:

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

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

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

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

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

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

INT_EXACTNESS, is an executable FORTRAN90 program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

INT_EXACTNESS_HERMITE, is an executable FORTRAN90 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 an executable FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.

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

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

PRODUCT_FACTOR is an executable FORTRAN90 program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is an executable FORTRAN90 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 FORTRAN90 library which contains 1-dimensional quadrature rules.

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 FORTRAN90 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.f90, the source code.
hermite_rule.csh, commands to compile the source code.

Examples and Tests:

output_cpp.txt, printed output from the command


            hermite_rule 4 0 C++

output_f77.txt, printed output from the command


            hermite_rule 4 0 F77

output_f90.txt, printed output from the command


            hermite_rule 4 0 F90

output_f90_modified.txt, printed output from the command


            hermite_rule 4 1 F90

output_mat.txt, printed output from the command


            hermite_rule 8 0 MAT

herm_o4_r.txt, the region file created by the command


            hermite_rule 4 0 herm_o4

herm_o4_w.txt, the weight file created by the command


            hermite_rule 4 0 herm_o4

herm_o4_x.txt, the abscissa file created by the command


            hermite_rule 4 0 herm_o4

herm_o4_modified_r.txt, the region file created by the command


            hermite_rule 4 1 herm_o4

herm_o4_modified_w.txt, the weight file created by the command


            hermite_rule 4 1 herm_o4

herm_o4_modified_x.txt, the abscissa file created by the command
```
            hermite_rule 4 1 herm_o4
          
```

List of Routines:

MAIN is the main program for HERMITE_RULE.
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.
DTABLE_CLOSE_WRITE closes a file used to write a DTABLE.
DTABLE_DATA_WRITE writes DTABLE data to a file.
DTABLE_HEADER_WRITE writes the header to a DTABLE file.
DTABLE_OPEN_WRITE opens a file to write a DTABLE.
DTABLE_WRITE writes a DTABLE to a file.
GET_UNIT returns a free FORTRAN unit number.
HERMITE_COMPUTE computes a Gauss-Hermite quadrature rule.
HERMITE_HANDLE tests computes the requested 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_GAMMA computes the gamma function.
R8_HUGE returns the largest legal R8.
S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
S_TO_I4 reads an I4 from a string.
S_TO_R8 reads an R8 from a string.
TIMESTAMP prints the current YMDHMS date as a time stamp.
TIMESTRING writes the current YMDHMS date into a string.

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

Last revised on 15 January 2008.

HERMITE_RULE Gauss-Hermite Quadrature Rules