QUADRATURE_RULES_GEGENBAUER is a dataset directory which contains examples of quadrature rules of Gauss-Gegenbauer type.

The Gauss-Gegenbauer quadrature rule is designed to approximate integrals on the interval [-1,1], with a weight function of the form (1-x^2)ALPHA. ALPHA is a real parameter that must be greater than -1.

Gauss-Gegenbauer quadrature assumes that the integrand we are considering has a form like:

```        Integral ( -1 <= x <= +1 ) (1-x^2)^alpha f(x) dx
```

The standard Gauss-Gegenbauer quadrature rule is used as follows:

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

For this directory, a quadrature rule is stored as three files, containing the weights, the points, and a file containing two points defining the endpoints of the region.

### Example:

We consider a Gauss-Gegenbauer quadrature rule of order 4 with ALPHA = 0.5.

Here is the text of the "W" file storing the weights of such a rule:

``````
0.2170787134227060
0.5683194499747424
0.5683194499747424
0.2170787134227060
``````

Here is the text of the "X" file storing the abscissas of such a rule:

``````
-0.8090169943749475
-0.3090169943749475
0.3090169943749474
0.8090169943749475
``````

Here is the text of the "R" file storing the lower and upper limits of the region:

``````
-1.0
+1.0
``````

### Related Data and Programs:

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

INT_EXACTNESS_GEGENBAUER, a FORTRAN90 program which reads files defining a Gauss-Gegenbauer quadrature rule, and tests it for exactness against monomial integrands.

### Sample Files:

Gauss-Gegenbauer Rule, Order 1, ALPHA = 0.5.

Gauss-Gegenbauer Rule, Order 2, ALPHA = 0.5.

Gauss-Gegenbauer Rule, Order 4, ALPHA = 0.5.

Gauss-Gegenbauer Rule, Order 8, ALPHA = 0.5.

Gauss-Gegenbauer Rule, Order 16, ALPHA = 0.5.

Gauss-Gegenbauer Rule, Order 32, ALPHA = 0.5.

You can go up one level to the DATASETS page.

Last revised on 25 February 2008.