# INT_EXACTNESS_GEN_HERMITE Exactness of Generalized Gauss-Hermite Quadrature Rules

INT_EXACTNESS_GEN_HERMITE is a FORTRAN90 program which investigates the polynomial exactness of a generalized Gauss-Hermite quadrature rule for the infinite interval (-oo,+oo).

Note that the arithmetic accuracy of this program may be insufficient to discriminant the accuracy of generalized Gauss-Hermite rules of moderate order - say, order 16. Therefore, a FORTRAN90 version of this program has been prepared, using "quadruple precision", which should give more satisfactory results for a wider range of orders. This program can be found cataloged below as INT_EXACTNESS_GEN_HERMITE_R16.

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) dx

where the factor |x|^alpha * exp(-x^2) is regarded as a weight 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) 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 |x|^alpha * exp(-x^2) is implicit. In that case, we are looking at approximating

Integral ( -oo < x < +oo ) f(x) dx

and 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) dx

may 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 divided by the weighting function evaluated at the corresponding abscissa.

For a standard generalized Gauss-Hermite rule, polynomial exactness is defined in terms of the function f(x). That is, we say the rule is exact for polynomials up to degree DEGREE_MAX if, for any polynomial f(x) of that degree or less, the quadrature rule will produce the exact value of

Integral ( -oo < x < +oo ) |x|^alpha * exp(-x^2) * f(x) dx

For a modified generalized Gauss-Hermite rule, polynomial exactness is defined in terms of the function f(x) divided by the implicit weighting function. That is, we say a modified generalized Gauss-Hermite rule is exact for polynomials up to degree DEGREE_MAX if, for any integrand f(x) with the property that f(x)/(|x|^alpha*exp(-x^2)) is a polynomial of degree no more than DEGREE_MAX, the quadrature rule will product the exact value of:

Integral ( -oo < x < +oo ) f(x) dx

The program starts at DEGREE = 0, and then proceeds to DEGREE = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of DEGREE, the program generates the corresponding monomial term, 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.

If the program understands that the rule being considered is a modified rule, then the monomials are multiplied by |x|^alpha * exp(-x^2) when performing the exactness test.

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.

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

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

• prefix_exact.txt

### Usage:

int_exactness_gen_hermite prefix degree_max alpha option
where
• prefix is 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.
• alpha is the value of the parameter, which should be a real number greater than -1.0. Setting alpha to 0.0 results in the basic (non-generalized) Gauss-Hermite rule.
• option:
0 indicates a standard rule for integrating |x|^alpha * exp(-x^2)*f(x).
1 indicates a modified rule for integrating f(x).

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

### Licensing:

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

### Languages:

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

### Related Data and Programs:

GEN_HERMITE_RULE, a FORTRAN90 program which can generate a generalized Gauss-Hermite quadrature rule on request.

HERMITE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

INT_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of a quadrature rule for a finite interval.

INT_EXACTNESS_CHEBYSHEV1, a FORTRAN90 program which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.

INT_EXACTNESS_CHEBYSHEV2, a FORTRAN90 program which tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.

INT_EXACTNESS_GEGENBAUER, a FORTRAN90 program which tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.

INT_EXACTNESS_GEN_LAGUERRE, a FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.

INT_EXACTNESS_JACOBI, a FORTRAN90 program which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

LAGUERRE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of Gauss-Laguerre quadrature rules for integration over [0,+oo) with density function exp(-x).

LEGENDRE_EXACTNESS, a FORTRAN90 program which tests the monomial exactness of quadrature rules for the Legendre problem of integrating a function with density 1 over the interval [-1,+1].

### Reference:

1. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
2. Arthur Stroud, Don Secrest,
Gaussian Quadrature Formulas,
Prentice Hall, 1966,
LC: QA299.4G3S7.

### Examples and Tests:

GEN_HERM_O1_A1.0 is a standard generalized Gauss-Hermite order 1 rule with ALPHA = 1.0.

GEN_HERM_O2_A1.0 is a standard generalized Gauss-Hermite order 2 rule with ALPHA = 1.0.

GEN_HERM_O4_A1.0 is a standard generalized Gauss-Hermite order 4 rule with ALPHA = 1.0.

GEN_HERM_O8_A1.0 is a standard generalized Gauss-Hermite order 8 rule with ALPHA = 1.0.

GEN_HERM_O16_A1.0 is a standard generalized Gauss-Hermite order 16 rule with ALPHA = 1.0.

GEN_HERM_O1_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 1 rule with ALPHA = 1.0.

GEN_HERM_O2_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 2 rule with ALPHA = 1.0.

GEN_HERM_O4_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 4 rule with ALPHA = 1.0.

GEN_HERM_O8_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 8 rule with ALPHA = 1.0.

GEN_HERM_O16_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 16 rule with ALPHA = 1.0.

### List of Routines:

• MAIN is the main program for INT_EXACTNESS_GEN_HERMITE.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_ROW_COUNT counts the number of row records in a file.
• GEN_HERMITE_INTEGRAL evaluates a monomial generalized Hermite integral.
• GET_UNIT returns a free FORTRAN unit number.
• MONOMIAL_QUADRATURE_GEN_HERMITE applies a quadrature rule to a monomial.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8MAT_DATA_READ reads data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.

Last revised on 28 January 2010.