Quadrature Rules

Generalized Gauss-Laguerre Type

**QUADRATURE_RULES_GEN_LAGUERRE**
is a dataset directory which
contains examples of quadrature rules of Generalized Gauss-Laguerre type.

Generalized Gauss-Legendre quadrature rules are a generalization of the
basic Gauss-Legendre quadrature rule. The generalization involves
including a factor of **x ^{alpha}** in the weight factor.
(The parameter

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

Integral ( A <= x < oo ) xwhere the factor^{alpha}* exp(-x) * f(x) dx

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

Integral ( A <= x < oo ) xmay be approximated by^{alpha}* exp(-x) * f(x) dx

Sum ( 1 <= I <= N ) w(i) * f(x(i))

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

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

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

Integral ( A <= x < oo ) f(x) dxmay be approximated by

Sum ( 1 <= I <= N ) w(i) * f(x(i))

When using a generalized 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 the inverse of the weight function 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.

We consider a standard generalized Gauss-Laguerre quadrature rule of order 4, with ALPHA = 0.5.

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

```
0.4530087825780014
0.3816169905147275
0.5079463161099657E-01
0.8065912142432840E-03
```

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

```
0.5235260767382691
2.156648763269094
5.137387546176711
10.18243761381592
```

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

```
0.0
1.0E+30
```

The modified rule of order 4 and ALPHA = 0.5 has the same values of X and R, but the modified weights are now:

```
1.056822251392305
2.245735732700531
3.815800963697504
6.681963149567727
```

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

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

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

Standard Generalized Gauss-Laguerre Rule, Order 4, ALPHA = -0.5:

- gen_lag_o4_am0.5_x.txt, the abscissas for the order 4 rule.
- gen_lag_o4_am0.5_w.txt, the weights.
- gen_lag_o4_am0.5_r.txt, the range of the integration region.

Standard Generalized Gauss-Laguerre Rule, Order 4, ALPHA = 0.0:

- gen_lag_o4_a0.0_x.txt, the abscissas for the order 4 rule.
- gen_lag_o4_a0.0_w.txt, the weights.
- gen_lag_o4_a0.0_r.txt, the range of the integration region.

Standard Generalized Gauss-Laguerre Rule, Order 1, ALPHA = 0.5:

- gen_lag_o1_a0.5_x.txt, the abscissas for the order 1 rule.
- gen_lag_o1_a0.5_w.txt, the weights.
- gen_lag_o1_a0.5_r.txt, the range of the integration region.

Standard Generalized Gauss-Laguerre Rule, Order 2, ALPHA = 0.5:

- gen_lag_o2_a0.5_x.txt, the abscissas for the order 4 rule.
- gen_lag_o2_a0.5_w.txt, the weights.
- gen_lag_o2_a0.5_r.txt, the range of the integration region.

Standard Generalized Gauss-Laguerre Rule, Order 4, ALPHA = 0.5:

- gen_lag_o4_a0.5_x.txt, the abscissas for the order 4 rule.
- gen_lag_o4_a0.5_w.txt, the weights.
- gen_lag_o4_a0.5_r.txt, the range of the integration region.

Standard Generalized Gauss-Laguerre Rule, Order 8, ALPHA = 0.5:

- gen_lag_o8_a0.5_x.txt, the abscissas for the order 8 rule.
- gen_lag_o8_a0.5_w.txt, the weights.
- gen_lag_o8_a0.5_r.txt, the range of the integration region.

Standard Generalized Gauss-Laguerre Rule, Order 16, ALPHA = 0.5:

- gen_lag_o16_a0.5_x.txt, the abscissas for the order 16 rule.
- gen_lag_o16_a0.5_w.txt, the weights.
- gen_lag_o16_a0.5_r.txt, the range of the integration region.

Standard Generalized Gauss-Laguerre Rule, Order 4, ALPHA = 1.0:

- gen_lag_o4_a1.0_x.txt, the abscissas for the order 4 rule.
- gen_lag_o4_a1.0_w.txt, the weights.
- gen_lag_o4_a1.0_r.txt, the range of the integration region.

Standard Generalized Gauss-Laguerre Rule, Order 4, ALPHA = 2.0:

- gen_lag_o4_a2.0_x.txt, the abscissas for the order 4 rule.
- gen_lag_o4_a2.0_w.txt, the weights.
- gen_lag_o4_a2.0_r.txt, the range of the integration region.

Standard Generalized Gauss-Laguerre Rule, Order 4, ALPHA = 3.0:

- gen_lag_o4_a3.0_x.txt, the abscissas for the order 4 rule.
- gen_lag_o4_a3.0_w.txt, the weights.
- gen_lag_o4_a3.0_r.txt, the range of the integration region.

Modified Generalized Gauss-Laguerre Rule, Order 4, ALPHA = 0.0:

- gen_lag_o4_a0.0_modified_x.txt, the abscissas for the order 4 rule.
- gen_lag_o4_a0.0_modified_w.txt, the weights.
- gen_lag_o4_a0.0_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Laguerre Rule, Order 1, ALPHA = 0.5:

- gen_lag_o1_a0.5_modified_x.txt, the abscissas for the order 1 rule.
- gen_lag_o1_a0.5_modified_w.txt, the weights.
- gen_lag_o1_a0.5_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Laguerre Rule, Order 2, ALPHA = 0.5:

- gen_lag_o2_a0.5_modified_x.txt, the abscissas for the order 2 rule.
- gen_lag_o2_a0.5_modified_w.txt, the weights.
- gen_lag_o2_a0.5_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Laguerre Rule, Order 4, ALPHA = 0.5:

- gen_lag_o4_a0.5_modified_x.txt, the abscissas for the order 4 rule.
- gen_lag_o4_a0.5_modified_w.txt, the weights.
- gen_lag_o4_a0.5_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Laguerre Rule, Order 8, ALPHA = 0.5:

- gen_lag_o8_a0.5_modified_x.txt, the abscissas for the order 8 rule.
- gen_lag_o8_a0.5_modified_w.txt, the weights.
- gen_lag_o8_a0.5_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Laguerre Rule, Order 16, ALPHA = 0.5:

- gen_lag_o16_a0.5_modified_x.txt, the abscissas for the order 16 rule.
- gen_lag_o16_a0.5_modified_w.txt, the weights.
- gen_lag_o16_a0.5_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Laguerre Rule, Order 4, ALPHA = 1.0:

- gen_lag_o4_a1.0_modified_x.txt, the abscissas for the order 4 rule.
- gen_lag_o4_a1.0_modified_w.txt, the weights.
- gen_lag_o4_a1.0_modified_r.txt, the range of the integration region.

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