# GEN_HERMITE_RULE Generalized Gauss-Hermite Quadrature Rules

GEN_HERMITE_RULE is a C++ 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:

CCN_RULE, a C++ program which defines a nested Clenshaw Curtis quadrature rule.

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

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

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

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

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

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

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

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

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

LATTICE_RULE, a C++ library which approximates M-dimensional integrals using lattice rules.

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

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

PATTERSON_RULE, a C++ 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.

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

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

TRUNCATED_NORMAL_RULE, a C++ 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
```

### List of Routines:

• MAIN is the main program for GEN_HERMITE_RULE.
• CDGQF computes a Gauss quadrature formula with default A, B and simple knots.
• CGQF computes knots and weights of a Gauss quadrature formula.
• CLASS_MATRIX computes the Jacobi matrix for a quadrature rule.
• IMTQLX diagonalizes a symmetric tridiagonal matrix.
• PARCHK checks parameters ALPHA and BETA for classical weight functions.
• R8_ABS returns the absolute value of an R8.
• R8_EPSILON returns the R8 roundoff unit.
• R8_HUGE returns a "huge" R8.
• R8_SIGN returns the sign of an R8.
• R8MAT_WRITE writes an R8MAT file with no header.
• RULE_WRITE writes a quadrature rule to three files.
• SCQF scales a quadrature formula to a nonstandard interval.
• SGQF computes knots and weights of a Gauss Quadrature formula.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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

Last revised on 23 February 2010.