QUADRATURE_RULES_CHEBYSHEV1 is a dataset directory which contains some examples of quadrature rules of Gauss-Chebyshev type 1.

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

Gauss-Chebyshev type 1 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 1 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 1 quadrature rule of order 4.

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

``````
0.7853981633974483
0.7853981633974483
0.7853981633974483
0.7853981633974483
``````

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

``````
-0.9238795325112867
-0.3826834323650898
0.3826834323650897
0.9238795325112867
``````

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:

CHEBYSHEV1_RULE, a C++ program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

INT_EXACTNESS_CHEBYSHEV1, a FORTRAN90 program which reads files defining a Gauss-Chebyshev type 1 quadrature rule, and test it for exactness against monomial integrands.

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

### Sample Files:

Gauss-Chebyshev Type 1 Rule, Order 1.

Gauss-Chebyshev Type 1 Rule, Order 2.

Gauss-Chebyshev Type 1 Rule, Order 4.

Gauss-Chebyshev Type 1 Rule, Order 8.

Gauss-Chebyshev Type 1 Rule, Order 16.

Gauss-Chebyshev Type 1 Rule, Order 32.

You can go up one level to the DATASETS page.

Last revised on 25 February 2008.