QUADRATURE_RULES_LAGUERRE is a dataset directory which contains examples of quadrature rules of Gauss-Laguerre type.

Gauss-Laguerre quadrature rules are designed to approximate integrals on semi-infinite intervals. By default, this interval is [0,+oo) but in some cases, the interval may be specified with a different initial point, which we will suggest by [A,+oo).

In the following discussion, we will represent the left endpoint by A, while understanding that it is usually taken to be 0.

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

```        Integral ( A <= x < +oo ) exp(-x) f(x) dx
```
where the factor exp(-x) is regarded as a weight factor.

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

```        Integral ( A <= x < +oo ) exp(-x) f(x) dx
```
may be approximated by
```        Sum ( 1 <= I <= N ) w(i) * f(x(i))
```

It is often convenient to consider approximating integrals in which the weighting factor exp(-x) is implicit. In that case, we are looking at approximating

```        Integral ( A <= x < +oo ) f(x) dx
```
and it is easy to modify a standard Gauss-Laguerre quadrature rule to handle this case directly.

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

```        Integral ( A <= x < +oo ) f(x) dx
```
may be approximated by
```        Sum ( 1 <= I <= N ) w(i) * f(x(i))
```

When using a Gauss-Laguerre quadrature rule, it's important to know whether the rule has been developed for the standard or modified cases. Basically, the only change is that the weights of the modified rule have been multiplied by an exponential factor evaluated at the corresponding abscissa.

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. The first endpoint is either 0 or A; the second endpoint is really infinity, and we will simply place a "large" value here, trusting that the user will already know that this is merely a placeholder for infinity.

### Example:

We consider a modified Gauss-Laguerre quadrature rule of order 4.

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

``````
0.60315
0.35741
0.03888
0.00053
``````
(We have reduced the number of digits printed in this presentation)

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

``````
0.32254
1.74576
4.53662
9.39507
``````

Here is the text of the "R" file storing the lower and upper limits of the region:

``````
0.0
1.0E+30
``````

### Related Data and Programs:

INT_EXACTNESS_LAGUERRE, a C++ program which tests the polynomial exactness of Gauss-Laguerre quadrature rules.

LAGUERRE_POLYNOMIAL, a C++ library which evaluates the Laguerre polynomial, the generalized Laguerre polynomial, and the Laguerre function.

LAGUERRE_RULE, a C++ program which can compute and print a Gauss-Laguerre quadrature rule.

QUADRATURE_RULES_GEN_LAGUERRE, a dataset directory which contains quadrature rules for integration on a semi-infinite interval, using a generalized Gauss-Laguerre rule.

TEST_INT_LAGUERRE, a C++ library which defines test integrands for integration over [A,+oo).

### Sample Files:

Standard Gauss-Laguerre Rules, Order 1 through 10, 16, 32, 64:

Modified Gauss-Laguerre Rules, Order 1, 2, 4, 8, 16, 32, 64:

You can go up one level to the DATASETS page.

Last revised on 13 May 2014.