QUADRATURE_RULES_PATTERSON is a dataset directory which contains examples of quadrature rules of Gauss-Patterson type.

Gauss-Patterson quadrature rules are designed to approximate integrals on the interval [-1,1].

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

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

The Gauss-Patterson quadrature is a nested family which begins with the Gauss-Legendre rules of orders 1 and 3, and then succesively inserts one new abscissa in each subinterval. Thus, after the second rule, the Gauss-Patterson rules do not have the super-high precision of the Gauss-Legendre rules. They trade this precision in exchange for the advantages of nestedness. This means that Gauss-Patterson rules are only available for orders of 1, 3, 7, 15, 31, 63, and 127.

A standard Gauss-Patterson quadrature rule is a set of n positive weights w and abscissas x so that

```        Integral ( -1 <= x <= +1 ) f(x) dx
```
may be approximated by
```        Sum ( 1 <= I <= N ) 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 standard Gauss-Patterson quadrature rule of order 7.

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

``````
0.1046562260264673
0.2684880898683334
0.4013974147759622
0.4509165386584741
0.4013974147759622
0.2684880898683334
0.1046562260264673
``````

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

``````
-0.9604912687080203
-0.7745966692414834
-0.4342437493468025
0.000000000000000
0.4342437493468025
0.7745966692414834
0.9604912687080203
``````

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:

PATTERSON_RULE, a C++ program which computes a 1D Gauss-Patterson quadrature rule.

### Sample Files:

Standard Gauss-Patterson Rule, Order 1:

Standard Gauss-Patterson Rule, Order 3:

Standard Gauss-Patterson Rule, Order 7:

Standard Gauss-Patterson Rule, Order 15:

Standard Gauss-Patterson Rule, Order 31:

Standard Gauss-Patterson Rule, Order 63:

Standard Gauss-Patterson Rule, Order 127:

You can go up one level to the DATASETS page.

Last revised on 21 January 2009.