GEN_HERMITE_RULE_SS is a C++ program which generates a specific generalized 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.

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.

Note that the program INT_EXACTNESS_GEN_HERMITE is provided to assess the accuracy of quadrature rules created by this program. However, for moderately large orders such as 16, that program may not produce very satisfactory results. For this reason, a FORTRAN90 version has been provided, named INT_EXACTNESS_GEN_HERMITE_R16, which uses "quadruple precision", and which should provide satisfactory results for a wider range of orders.

Usage:

gen_hermite_rule_ss order alpha option output
where
• order is the number of points in the quadrature rule. A typical value might be 4, 8, or 16.
• 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.
• option specifies the rule:
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.

Languages:

GEN_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 available in a C++ version and a FORTRAN90 version and a MATLAB version

GEN_LAGUERRE_RULE, is a C++ program which computes a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, is a C++ program which computes a generalized Gauss-Hermite quadrature rule.

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

INT_EXACTNESS_GEN_HERMITE_R16 is a FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules. It is a version of INT_EXACTNESS_GEN_HERMITE that uses "double double precision" real arithmetic, and gives much improved results.

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

JACOBI_RULE, is a C++ program which computes a generalized Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, is a C++ program which computes 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 a C++ program which constructs a product rule from distinct 1D factor rules.

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

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

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

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

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

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

Examples and Tests:

• output_cpp.txt, printed output from the command
```
gen_hermite_rule_ss 4 1.0 0 C++
```
• output_f77.txt, printed output from the command
```
gen_hermite_rule_ss 4 -0.5 0 F77
```
• output_f90.txt, printed output from the command
```
gen_hermite_rule_ss 4 0.0 0 F90
```
• output_f90_modified.txt, printed output from the command
```
gen_hermite_rule_ss 4 0.0 1 F90
```
• output_mat.txt, printed output from the command
```
gen_hermite_rule_ss 8 3.0 0 MAT
```
• gen_herm_o4_a1.0_r.txt, the region file created by the command
```
gen_hermite_rule_ss 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_ss 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_ss 4 1.0 0 gen_herm_o4_a1.0
```

List of Routines:

• MAIN is the main program for GEN_HERMITE_RULE.
• GEN_HERMITE_COMPUTE computes a generalized Gauss-Hermite rule.
• GEN_HERMITE_HANDLE computes a generalized Gauss-Hermite 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 a root of a generalized Laguerre 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.
• R8MAT_WRITE writes an R8MAT file with no header.
• 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.