# GEN_HERMITE_RULE Generalized Gauss-Hermite Quadrature Rules

GEN_HERMITE_RULE is a MATLAB program which generates a specific generalized Gauss-Hermite quadrature rule, based on user input.

The rule is written to three files for easy use as input to other programs.

The generalized Gauss Hermite quadrature rule is used as follows:

```        Integral ( -oo < x < +oo ) |x-a|^alpha * exp( - b * ( x - a)^2 ) f(x) dx
```
is to be approximated by
```        Sum ( 1 <= i <= order ) w(i) * f(x(i))
```

### Usage:

gen_hermite_rule ( order, alpha, a, b, 'filename' )
where
• order is the number of points in the quadrature rule.
• 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.
• a is the center point (default 0);
• b is the scale factor (default 1);
• 'filename' specifies the names of the output files: filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

### Languages:

GEN_HERMITE_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version

### Related Data and Programs:

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

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

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

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

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

GEN_LAGUERRE_RULE, a MATLAB program which computes a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a MATLAB program which computes a generalized Gauss-Hermite quadrature rule.

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

JACOBI_RULE, a MATLAB program which computes a generalized Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a MATLAB program which computes a Gauss-Laguerre quadrature rule.

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

LINE_FELIPPA_RULE, a MATLAB library which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

PATTERSON_RULE, a MATLAB program which computes a Gauss-Patterson quadrature rule.

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

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

TRUNCATED_NORMAL_RULE, a MATLAB program which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian distribution, that has been truncated to [A,+oo), (-oo,B] or [A,B].

### 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. 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.
4. Jaroslav Kautsky, Sylvan Elhay,
Calculation of the Weights of Interpolatory Quadratures,
Numerische Mathematik,
Volume 40, 1982, pages 407-422.
5. Roger Martin, James Wilkinson,
The Implicit QL Algorithm,
Numerische Mathematik,
Volume 12, Number 5, December 1968, pages 377-383.
6. Arthur Stroud, Don Secrest,
Prentice Hall, 1966,
LC: QA299.4G3S7.

### Examples and Tests:

• gen_herm_o4_a1.0_r.txt, the region file created by the command
```
gen_hermite_rule ( 4, 1.0, 0.0, 1.0, '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, 0.0, 1.0, '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, 0.0, 1.0, 'gen_herm_o4_a1.0' )
```

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

Last revised on 24 February 2010.