# gegenbauer_rule

gegenbauer_rule, a MATLAB code which generates a specific Gauss-Gegenbauer quadrature rule, based on user input.

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

The Gauss-Gegenbauer quadrature rule is used as follows:

```        Integral ( A <= x <= B ) ((x-A)(B-X))^alpha f(x) dx
```
is to be approximated by
```        Sum ( 1 <= i <= order ) w(i) * f(x(i))
```
where alpha is a real parameter greater than -1.

### Usage:

gegenbauer_rule ( order, alpha, a, b, 'filename' )
where
• order is the number of points in the quadrature rule.
• alpha is the exponent of (1-x^2), which must be greater than -1.
• a is the left endpoint;
• b is the right endpoint.
• 'filename' specifies the filenames: filename_w.txt, filename_x.txt, and filename_r.txt, containing the weights, abscissas, and interval limits.

### Languages:

gegenbauer_rule is available in a C++ version and a FORTRAN90 version and a MATLAB version.

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quadrature_rules_gegenbauer, a dataset directory which contains triples of files defining Gauss-Gegenbauer quadrature rules.

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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:

Last revised on 24 January 2019.