Quadrature Rules of Generalized Gauss-Hermite Type

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

Generalized Gauss-Hermite quadrature rules are a generalization of the
basic Gauss-Hermite quadrature rule. The generalization involves
including a factor of **|x|^alpha** in the weight factor.
(The parameter **alpha** is a real value which may take on any value
greater than -1.0.)

Generalized Gauss-Hermite quadrature rules are designed to approximate integrals on infinite intervals.

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

Integral ( -oo < x < oo ) |x|^alpha * exp(-x^2) * f(x) dxwhere the factor

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

Integral ( -oo < x < oo ) |x|^alpha * exp(-x^2) * f(x) dxmay 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 **|x|^alpha * exp(-x^2)** is implicit.
In that case, we are looking at approximating

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

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

Integral ( -oo < x < oo ) f(x) dxmay be approximated by

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

When using a generalized Gauss-Hermite 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-Hermite quadrature rule of order 4, with ALPHA = 1.0.

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

```
0.7322330470336313E-01
0.4267766952966369
0.4267766952966369
0.7322330470336313E-01
```

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

```
-1.847759065022573
-0.7653668647301796
0.7653668647301796
1.847759065022573
```

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

```
-1.0E+30
1.0E+30
```

The modified rule of order 4 and ALPHA = 1.0 has the same values of X and R, but the modified weights are now:

```
1.204420375824328
1.001693503977319
1.001693503977319
1.204420375824328
```

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

GEN_HERMITE_RULE, a FORTRAN90 program which can compute a generalized Gauss-Hermite quadrature rule (standard or modified) and print it, or save it as weight, abscissa, and region files.

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

QUADRATURE_RULES_HERMITE_PHYSICIST, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2).

QUADRATURE_RULES_HERMITE_PROBABILIST, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function exp(-x^2/2).

QUADRATURE_RULES_HERMITE_UNWEIGHTED, a dataset directory which contains Gauss-Hermite quadrature rules, for integration on the interval (-oo,+oo), with weight function 1.

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.

QUADRULE, a FORTRAN90 library which defines various quadrature rules.

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

- gen_herm_o4_am0.5_x.txt, the abscissas for the order 4 rule.
- gen_herm_o4_am0.5_w.txt, the weights.
- gen_herm_o4_am0.5_r.txt, the range of the integration region.

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

- gen_herm_o4_a0.0_x.txt, the abscissas for the order 4 rule.
- gen_herm_o4_a0.0_w.txt, the weights.
- gen_herm_o4_a0.0_r.txt, the range of the integration region.

Standard Generalized Gauss-Hermite Rule, Order 1, ALPHA = 1.0:

- gen_herm_o1_a1.0_x.txt, the abscissas for the order 1 rule.
- gen_herm_o1_a1.0_w.txt, the weights.
- gen_herm_o1_a1.0_r.txt, the range of the integration region.

Standard Generalized Gauss-Hermite Rule, Order 2, ALPHA = 1.0:

- gen_herm_o2_a1.0_x.txt, the abscissas for the order 4 rule.
- gen_herm_o2_a1.0_w.txt, the weights.
- gen_herm_o2_a1.0_r.txt, the range of the integration region.

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

- gen_herm_o4_a1.0_x.txt, the abscissas for the order 4 rule.
- gen_herm_o4_a1.0_w.txt, the weights.
- gen_herm_o4_a1.0_r.txt, the range of the integration region.

Standard Generalized Gauss-Hermite Rule, Order 8, ALPHA = 1.0:

- gen_herm_o8_a1.0_x.txt, the abscissas for the order 8 rule.
- gen_herm_o8_a1.0_w.txt, the weights.
- gen_herm_o8_a1.0_r.txt, the range of the integration region.

Standard Generalized Gauss-Hermite Rule, Order 16, ALPHA = 1.0:

- gen_herm_o16_a1.0_x.txt, the abscissas for the order 16 rule.
- gen_herm_o16_a1.0_w.txt, the weights.
- gen_herm_o16_a1.0_r.txt, the range of the integration region.

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

- gen_herm_o4_a2.0_x.txt, the abscissas for the order 4 rule.
- gen_herm_o4_a2.0_w.txt, the weights.
- gen_herm_o4_a2.0_r.txt, the range of the integration region.

Standard Generalized Gauss-Hermite Rule, Order 4, ALPHA = 4.0:

- gen_herm_o4_a4.0_x.txt, the abscissas for the order 4 rule.
- gen_herm_o4_a4.0_w.txt, the weights.
- gen_herm_o4_a4.0_r.txt, the range of the integration region.

Standard Generalized Gauss-Hermite Rule, Order 4, ALPHA = 6.0:

- gen_herm_o4_a6.0_x.txt, the abscissas for the order 4 rule.
- gen_herm_o4_a6.0_w.txt, the weights.
- gen_herm_o4_a6.0_r.txt, the range of the integration region.

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

- gen_herm_o4_a0.0_modified_x.txt, the abscissas for the order 4 rule.
- gen_herm_o4_a0.0_modified_w.txt, the weights.
- gen_herm_o4_a0.0_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Hermite Rule, Order 1, ALPHA = 1.0:

- gen_herm_o1_a1.0_modified_x.txt, the abscissas for the order 1 rule.
- gen_herm_o1_a1.0_modified_w.txt, the weights.
- gen_herm_o1_a1.0_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Hermite Rule, Order 2, ALPHA = 1.0:

- gen_herm_o2_a1.0_modified_x.txt, the abscissas for the order 2 rule.
- gen_herm_o2_a1.0_modified_w.txt, the weights.
- gen_herm_o2_a1.0_modified_r.txt, the range of the integration region.

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

- gen_herm_o4_a1.0_modified_x.txt, the abscissas for the order 4 rule.
- gen_herm_o4_a1.0_modified_w.txt, the weights.
- gen_herm_o4_a1.0_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Hermite Rule, Order 8, ALPHA = 1.0:

- gen_herm_o8_a1.0_modified_x.txt, the abscissas for the order 8 rule.
- gen_herm_o8_a1.0_modified_w.txt, the weights.
- gen_herm_o8_a1.0_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Hermite Rule, Order 16, ALPHA = 1.0:

- gen_herm_o16_a1.0_modified_x.txt, the abscissas for the order 16 rule.
- gen_herm_o16_a1.0_modified_w.txt, the weights.
- gen_herm_o16_a1.0_modified_r.txt, the range of the integration region.

Modified Generalized Gauss-Hermite Rule, Order 4, ALPHA = 2.0:

- gen_herm_o4_a2.0_modified_x.txt, the abscissas for the order 4 rule.
- gen_herm_o4_a2.0_modified_w.txt, the weights.
- gen_herm_o4_a2.0_modified_r.txt, the range of the integration region.

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