GEN_HERMITE_RULE_R16
Generalized Gauss-Hermite Quadrature Rules

GEN_HERMITE_RULE_R16 is an interactive FORTRAN90 program, using "double double precision" real arithmetic, which can generate a specific generalized Gauss-Hermite quadrature rule, based on user input.

This program is a fairly simple minded translation of GEN_HERMITE_RULE, in which variables declared REAL(KIND=8) became REAL(KIND=16). Almost no other attempt was made to improved accuracy. For instance, the routine computing the Gamma function is the same in both programs, whereas a more accurate routine might be suitable for the double double precision code.

This program was written to try to improve the results that were achieved when quadrature files generated by GEN_HERMITE_RULE were "diagnosed" by INT_EXACTNESS_GEN_HERMITE. In fact, using this program did not make a great deal of difference. Instead, creating a double double precision version of the diagnosis routine, called INT_EXACTNESS_GEN_HERMITE_R16 did the trick. It seems that in this case the doctor was sicker than the patient! Nonetheless, we are keeping this program around for future improvement and investigation.

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.

Both the basic and generalized Gauss-Hermite quadrature rules are designed to approximate integrals on infinite intervals.

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

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

where the factor |x|^alpha * exp(-x^2) is regarded as a weight factor.

The standard generalized Gauss Hermite quadrature rule is used as follows:

        Integral ( -oo < x < +oo ) |x|^alpha * exp(-x^2) 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 |x|^alpha * exp(-x^2) 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 generalized 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))

Note that when the real paramater alpha is equal to 0.0, we recover the original (that is, "non-generalized") Gauss-Hermite quadrature rule. Also note that the generalized Gauss-Hermite rule is only defined for parameter values alpha which are greater than -1.0.

Usage:

gen_hermite_rule_r16 order alpha option output

order

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

alpha

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.

option

a 0 value requests the "standard" integration rule for

            Integral ( -oo < x < +oo ) |x|^ALPHA * exp(-x^2) 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:

GEN_HERMITE_RULE is an executable FORTRAN90 program which computes a generalized Gauss-Hermite rule.

GEN_LAGUERRE_RULE, is an executable FORTRAN90 program which computes a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, is an executable FORTRAN90 program which computes a generalized Gauss-Hermite quadrature rule.

INT_EXACTNESS_GEN_HERMITE, is an executable FORTRAN90 program which checks the polynomial exactness of a generalized Gauss-Hermite quadrature rule.

INT_EXACTNESS_GEN_HERMITE_R16, is an executable FORTRAN90 program which checks the polynomial exactness of a generalized Gauss-Hermite quadrature rule, using "double double precision" real arithmetic.

INTLIB is a FORTRAN90 library containing a variety of routines for numerical estimation of integrals in 1D.

JACOBI_RULE, is an executable FORTRAN90 program which computes a generalized Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, is an executable FORTRAN90 program which computes 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 for constructing a product rule from identical 1D factor rules.

QUADPACK is a FORTRAN90 library containing a variety of routines for numerical estimation of integrals in 1D.

QUADRATURE_RULES_GEN_HERMITE is a dataset directory of triples of files defining generalized Gauss-Hermite quadrature rules.

QUADRULE is a FORTRAN90 library containing 1-dimensional quadrature rules.

TEST_INT_HERMITE is a FORTRAN90 library which defines test integrands that can be approximated using a Gauss-Hermite rule.

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:

gen_hermite_rule_r16.f90, the source code.
gen_hermite_rule_r16.csh, commands to compile the source code.

Examples and Tests:

output_cpp.txt, printed output from the command


            gen_hermite_rule 4 1.0 0 C++

output_f77.txt, printed output from the command


            gen_hermite_rule 4 -0.5 0 F77

output_f90.txt, printed output from the command


            gen_hermite_rule 4 0.0 0 F90

output_f90_modified.txt, printed output from the command


            gen_hermite_rule 4 0.0 1 F90

output_mat.txt, printed output from the command


            gen_hermite_rule 8 3.0 0 MAT

gen_herm_o4_a1.0_r.txt, the region file created by the command


            gen_hermite_rule 4 1.0 0 gen_herm_o4_a1.0

gen_herm_o4_a1.0_w.txt, the weight file created by the command


            gen_hermite_rule 4 1.0 0 gen_herm_o4_a1.0

gen_herm_o4_a1.0_x.txt, the abscissa file created by the command


            gen_hermite_rule 4 1.0 0 gen_herm_o4_a1.0

List of Routines:

MAIN is the main program for GEN_HERMITE_RULE_R16.
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.
GEN_HERMITE_COMPUTE computes a generalized Gauss-Hermite rule.
GEN_HERMITE_HANDLE computes the requested rule and outputs it.
GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre quadrature rule.
GEN_LAGUERRE_RECUR evaluates a generalized Laguerre polynomial.
GEN_LAGUERRE_ROOT improves an approximate root of a generalized Laguerre polynomial.
GET_UNIT returns a free FORTRAN unit number.
R16_GAMMA evaluates Gamma(X) for a real argument.
R16_HUGE returns a very large R16.
S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
S_TO_I4 reads an I4 from a string.
S_TO_R16 reads an R16 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 02 March 2008.

GEN_HERMITE_RULE_R16 Generalized Gauss-Hermite Quadrature Rules