QUADRATURE_RULES_LEGENDRE is a dataset directory which contains examples of quadrature rules of Gauss-Legendre type.

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

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

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

A standard Gauss-Legendre 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-Legendre quadrature rule of order 4.

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

``````
0.3478548451374538
0.6521451548625461
0.6521451548625461
0.3478548451374538
``````

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

``````
-0.8611363115940526
-0.3399810435848563
0.3399810435848563
0.8611363115940526
``````

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:

INT_EXACTNESS_LEGENDRE, a FORTRAN90 program which can read a set of files defining a Gauss-Legendre quadrature rule, and test it for exactness against monomial integrands.

LEGENDRE_RULE, a FORTRAN90 program which generates a Gauss-Legendre rule of the desired order.

QUADRATURE_TEST, a MATLAB program which reads the definition of a multidimensional quadrature rule from three files, applies the rule to a number of test integrals, and prints the results.

TEST_INT, a FORTRAN90 library which defines several test integrands which may be integrated.

### Sample Files:

Gauss-Legendre Rule, Order 1:

Gauss-Legendre Rule, Order 2:

Gauss-Legendre Rule, Order 3:

Gauss-Legendre Rule, Order 4:

Gauss-Legendre Rule, Order 7:

Gauss-Legendre Rule, Order 8:

Gauss-Legendre Rule, Order 15:

Gauss-Legendre Rule, Order 16:

Gauss-Legendre Rule, Order 31:

Gauss-Legendre Rule, Order 32:

Gauss-Legendre Rule, Order 63:

Gauss-Legendre Rule, Order 64:

Gauss-Legendre Rule, Order 65:

Gauss-Legendre Rule, Order 127:

You can go up one level to the DATASETS page.

Last revised on 19 February 2014.