# hermite_rule

hermite_rule, an Octave code which generates a specific Gauss-Hermite quadrature rule, based on user input.

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

The Gauss-Hermite quadrature rule is used as follows:

```        c * Integral ( -oo < x < +oo ) f(x) exp ( - b * ( x - a )^2 ) dx
```
is to be approximated by
```        Sum ( 1 <= i <= order ) w(i) * f(x(i))
```
Generally, a Gauss-Hermite quadrature rule of n points will produce the exact integral when f(x) is a polynomial of degree 2n-1 or less.

The value of C in front of the integral depends on the user's choice of the SCALE parameter:

• scale=0, then C = 1; this is the standard choice for Gauss-Hermite quadrature.
• scale=1, then C is a normalization factor so that f(x)=1 will integrate to 1. This implies in turn that the weights will sum to 1. This choice is appropriate when using the rule to compute probabilities.

### Usage:

hermite_rule ( order, a, b, scale 'filename' )
where
• order is the number of points in the quadrature rule.
• a is the center point (default 0);
• b is the scale factor (default 1);
• scale is the normalization option (0/1). If 1, then the weights are normalized to have unit sum;
• 'filename' specifies the output filenames: filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

### Languages:

hermite_rule is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

### Related Data and Programs:

alpert_rule, an Octave code which can set up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

ccn_rule, an Octave code which defines a nested Clenshaw Curtis quadrature rule.

chebyshev1_rule, an Octave code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

chebyshev2_rule, an Octave code which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

clenshaw_curtis_rule, an Octave code which defines a Clenshaw Curtis quadrature rule.

gegenbauer_rule, an Octave code which can compute and print a Gauss-Gegenbauer quadrature rule.

gen_hermite_rule, an Octave code which can compute and print a generalized Gauss-Hermite quadrature rule.

gen_laguerre_rule, an Octave code which can compute and print a generalized Gauss-Laguerre quadrature rule.

hermite_exactness, an Octave code which tests the polynomial exactness of Gauss-Hermite quadrature rules.

hermite_polynomial, an Octave code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

jacobi_rule, an Octave code which can compute and print a Gauss-Jacobi quadrature rule.

laguerre_rule, an Octave code which can compute and print a Gauss-Laguerre quadrature rule.

legendre_rule, an Octave code which computes a Gauss-Legendre quadrature rule.

line_felippa_rule, an Octave code which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

patterson_rule, an Octave code which computes a Gauss-Patterson quadrature rule.

quadrature_rules_hermite_physicis, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2).

quadrature_rules_hermite_probabilist, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2/2).

quadrature_rules_hermite_unweighted, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function 1.

truncated_normal_rule, an Octave code 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,