# hermite_rule

hermite_rule, a Python code which returns Gauss-Hermite quadrature rule.

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 a Octave version and a Python version.

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### 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.

### Source Code:

HERM_O4 is a Hermite rule of order 4, created by the command

```        hermite_rule ( 4, 0.0, 1.0, 0, 'herm_o4' )
```

Last revised on 26 May 2023.