# INT_EXACTNESS_HERMITE Exactness of Gauss-Hermite Quadrature Rules

INT_EXACTNESS_HERMITE is a MATLAB program which investigates the polynomial exactness of a Gauss-Hermite quadrature rule for the infinite interval (-oo,+oo).

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

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

We consider three variations of the rule, depending on the form of the weight factor w(x):

• option = 0, the unweighted rule:
```            Integral ( -oo < x < +oo ) f(x) dx
```
• option = 1, the physicist weighted rule:
```            Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx
```
• option = 2, the probabilist weighted rule:
```            Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx
```

The corresponding Gauss-Hermite rule that uses order points will approximate the integral by

```        sum ( 1 <= i <= order ) w(i) * f(x(i))
```
where, confusingly, w(i) is a vector of quadrature weights, which has no connection with the w(x) weight function.

When using a Gauss-Hermite quadrature rule, it's important to know whether the rule has been developed for the unweighted, physicist weighted or probabilist weighted cases.

For an unweighted Gauss-Hermite rule, polynomial exactness may be defined by assuming that f(x) has the form f(x) = exp(-x*x) * x^n for some nonnegative integer exponent n. We say an unweighted Gauss-Hermite rule is exact for polynomials up to degree DEGREE_MAX if the quadrature rule will produce the correct value of the integrals of such integrands for all exponents n from 0 to DEGREE_MAX.

For a physicist or probabilist weighted Gauss-Hermite rules, polynomial exactness may be defined by assuming that f(x) has the form f(x) = x^n for some nonnegative integer exponent n. We say the physicist or probabilist weighted Gauss-Hermite rule is exact for polynomials up to degree DEGREE_MAX if the quadrature rule will produce the correct value of the integrals of such integrands for all exponents n from 0 to DEGREE_MAX.

To test the polynomial exactness of a Gauss-Hermite quadrature rule of one of these three forms, the program starts at n = 0, and then proceeds to n = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of n, the program generates the appropriate corresponding integrand function (either exp(-x*x)*x^n or x^n), applies the quadrature rule to it, and determines the quadrature error. The program uses a scaling factor on each monomial so that the exact integral should always be 1; therefore, each reported error can be compared on a fixed scale.

The program is very flexible and interactive. The quadrature rule is defined by three files, to be read at input, and the maximum degree top be checked is specified by the user as well.

Note that the three files that define the quadrature rule are assumed to have related names, of the form

• prefix_x.txt
• prefix_w.txt
• prefix_r.txt
When running the program, the user only enters the common prefix part of the file names, which is enough information for the program to find all three files.

Note that when approximating these kinds of integrals, or even when evaluating an exact formula for these integrals, numerical inaccuracies can become overwhelming. The formula for the exact integral of x^n*exp(-x*x) (which we use to test for polynomial exactness) involves the double factorial function, which "blows up" almost as fast as the ordinary factorial. Thus, even for formulas of order 16, where we would like to consider monomials up to degree 31, the evaluation of the exact formula loses significant accuracy.

For information on the form of these files, see the QUADRATURE_RULES_HERMITE directory listed below.

The exactness results are written to an output file with the corresponding name:

• prefix_exact.txt

### Usage:

int_exactness_hermite ( 'prefix', degree_max, option )
where
• 'prefix' is a quoted string containing the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;
• degree_max is the maximum monomial degree to check. This would normally be a relatively small nonnegative number, such as 5, 10 or 15.
• option:
0 indicates the unweighted rule for integrating f(x),
1 indicates the physicist weighted rule for integrating exp(-x*x)*f(x),
2 indicates the probabilist weighted rule for integrating exp(-x*x/2)*f(x).

If the arguments are not supplied on the command line, the program will prompt for them.

### Languages:

INT_EXACTNESS_HERMITE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

HERMITE_RULE, a MATLAB program which generates a Gauss-Hermite quadrature rule on request.

INT_EXACTNESS, a MATLAB program which tests the polynomial exactness of a quadrature rule for a finite interval.

INT_EXACTNESS_CHEBYSHEV1, a MATLAB program which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.

INT_EXACTNESS_CHEBYSHEV2, a MATLAB program which tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.

INT_EXACTNESS_GEGENBAUER, a MATLAB program which tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.

INT_EXACTNESS_GEN_HERMITE, a MATLAB program which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.

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

INT_EXACTNESS_JACOBI, a MATLAB program which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

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

INT_EXACTNESS_LEGENDRE, a MATLAB program which tests the polynomial exactness of Gauss-Legendre quadrature rules.

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.

TEST_INT_HERMITE, a MATLAB library which defines integrand functions that can be approximately integrated by a Gauss-Hermite rule.

### Reference:

1. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.

### Source Code:

You can go up one level to the MATLAB source codes.

Last revised on 18 November 2011.