QUADRATURE_RULES_HERMITE_PROBABILIST is a dataset directory which contains examples of quadrature rules of Gauss-Hermite type using the probabilist weight.

Gauss-Hermite quadrature rules are designed to approximate integrals on the infinite interval (-oo,+oo).

The Gauss Hermite quadrature assumes that the integrand we are considering has a form like:

```        Integral ( -oo < x < +oo ) f(x) * rho(x) dx
```
where the factor w(x) is regarded as a weight factor.

There are variations of the rule, depending on the form of the weight factor rho(x):

• option = 0, the unweighted rule:
```            rho(x) = 1
```
• option = 1, the physicist weighted rule:
```            rho = exp(-x*x)
```
• option = 2, the probabilist weighted rule:
```            rho = exp(-x*x/2)
```
• option = 3, the physicist normalized weighted rule:
```            rho = exp(-x*x) / sqrt ( pi )
```
• option = 4, the probabilist normalized weighted rule:
```            rho = exp(-x*x/2) / sqrt ( 2 pi )
```

The corresponding Gauss-Hermite rule that uses order points will approximate the integral 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. Since the Hermite rules are defined on an infinite region, we set the endpoints to a very large negative and positive values respectively, and hope the program will understand what we mean.

If a Gauss-Hermite rule of order n is computed and evaluated with no numerical error, then it should produce the exact integral of any (weighted) polynomial of degree 2*n-1 or less.

As a simple check, for the unnormalized probabilist rule, the weights should add to sqrt(2*pi) = 2.5066282746310005024, while for the normalized probabilist rule, the weights should add to 1.

### Example:

We consider a probabilist unnormalized weighted Gauss-Hermite quadrature rule of order 3.

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

``````
0.4177713791051669
1.671085516420666
0.4177713791051672
``````

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

``````
-1.732050807568878
-0.1394816914999491E-15
1.732050807568878
``````

Here is the text of the "R" file storing the lower and upper limits of the region. These are formally -oo and +oo, but here we simply give them as large values.

``````
-0.1000000000000000E+31
0.1000000000000000E+31
``````

### Related Data and Programs:

HERMITE_POLYNOMIAL, a C++ library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

HERMITE_RULE, a C++ program which can compute and print a Gauss-Hermite quadrature rule.

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

QUADRATURE_RULES_GEN_HERMITE, a dataset directory which contains quadrature rules for integration on an infinite interval, using a generalized Gauss-Hermite rule.

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

SPARSE_GRID_HERMITE, a dataset directory which contains M-dimensional Smolyak sparse grids based on the 1D Gauss-Hermite rule;

TEST_INT_HERMITE, a C++ library which defines test integrands for integration over (-oo,+oo).

### Sample Files:

Gauss-Hermite Quadrature Rule, Probabilist Unnormalized Weight, Order 1:

Gauss-Hermite Quadrature Rule, Probabilist Unnormalized Weight, Order 3:

Gauss-Hermite Quadrature Rule, Probabilist Unnormalized Weight, Order 7:

Gauss-Hermite Quadrature Rule, Probabilist Normalized Weight, Order 10:

Gauss-Hermite Quadrature Rule, Probabilist Unnormalized Weight, Order 15:

Gauss-Hermite Quadrature Rule, Probabilist Unnormalized Weight, Order 31:

Gauss-Hermite Quadrature Rule, Probabilist Unnormalized Weight, Order 63:

Gauss-Hermite Quadrature Rule, Probabilist Unnormalized Weight, Order 127:

You can go up one level to the DATASETS page.

Last revised on 08 June 2013.