INT_EXACTNESS_GEN_HERMITE
Exactness of Generalized Gauss-Hermite Quadrature Rules


INT_EXACTNESS_GEN_HERMITE is a MATLAB 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

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:

Usage:

int_exactness_gen_hermite ( 'prefix', degree_max, alpha, option )
where

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 MATLAB program which can generate a generalized Gauss-Hermite quadrature rule on request.

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

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_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.

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

LEGENDRE_EXACTNESS, a MATLAB 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.

Source Code:

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.

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


Last revised on 05 August 2009.