Quadrature Rules of Gauss-Patterson Type

**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) dxmay 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.

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
```

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

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

Standard Gauss-Patterson Rule, Order 1:

- gp_o1_x.txt, the abscissas.
- gp_o1_w.txt, the weights.
- gp_o1_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 3:

- gp_o3_x.txt, the abscissas.
- gp_o3_w.txt, the weights.
- gp_o3_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 7:

- gp_o7_x.txt, the abscissas.
- gp_o7_w.txt, the weights.
- gp_o7_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 15:

- gp_o15_x.txt, the abscissas.
- gp_o15_w.txt, the weights.
- gp_o15_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 31:

- gp_o31_x.txt, the abscissas.
- gp_o31_w.txt, the weights.
- gp_o31_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 63:

- gp_o63_x.txt, the abscissas.
- gp_o63_w.txt, the weights.
- gp_o63_r.txt, the range of the integration region.

Standard Gauss-Patterson Rule, Order 127:

- gp_o127_x.txt, the abscissas.
- gp_o127_w.txt, the weights.
- gp_o127_r.txt, the range of the integration region.

You can go up one level to the DATASETS page.