QUADRATURE_RULES_CHEBYSHEV2 is a dataset directory which contains examples of quadrature rules of Gauss-Chebyshev type 2.

The Gauss-Chebyshev type 2 quadrature rule is designed to approximate integrals on the interval [-1,1], with a weight function of the form sqrt (1-x^2).

Gauss-Chebyshev type 2 quadrature assumes that the integrand we are considering has a form like:

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

The standard Gauss-Chebyshev type 2 quadrature rule is used as follows:

```        Integral ( -1 <= x <= +1 ) f(x) * sqrt(1-x^2) 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-Chebyshev type 2 quadrature rule of order 4.

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

``````
0.2170787134227061
0.5683194499747424
0.5683194499747423
0.2170787134227060
``````

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

``````
-0.8090169943749473
-0.3090169943749473
0.3090169943749475
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:

CHEBYSHEV2_RULE, a C++ program which computes and prints a Gauss-Chebyshev type 2 quadrature rule.

INT_EXACTNESS_CHEBYSHEV2, a FORTRAN90 program which reads files defining a Gauss-Chebyshev type 2 quadrature rule, and test it for exactness against monomial integrands.

QUADRATURE_RULES_CHEBYSHEV1, a dataset directory which contains quadrature rules for integration on [-1,+1], using a Gauss-Chebyshev type 1 rule.

TEST_INT, a C++ library which defines test integrands for 1D quadrature rules.

### Sample Files:

Gauss-Chebyshev Type 2 Rule, Order 1.

Gauss-Chebyshev Type 2 Rule, Order 2.

Gauss-Chebyshev Type 2 Rule, Order 4.

Gauss-Chebyshev Type 2 Rule, Order 8.

Gauss-Chebyshev Type 2 Rule, Order 16.

Gauss-Chebyshev Type 2 Rule, Order 32.

You can go up one level to the DATASETS page.

Last revised on 25 February 2008.