HERMITE_EXACTNESS
Exactness of Gauss-Hermite Quadrature Rules


HERMITE_EXACTNESS is a FORTRAN90 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 ) f(x) * rho(x) dx
      
where rho(x) is regarded as a weight factor.

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

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

        sum ( 1 <= i <= order ) w(i) * f(x(i))
      

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

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.

Usage:

hermite_exactness prefix degree_max 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:

HERMITE_EXACTNESS is available in a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CUBE_EXACTNESS, a FORTRAN90 library which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3D.

EXACTNESS, a FORTRAN90 library which investigates the exactness of quadrature rules that estimate the integral of a function with a density, such as 1, exp(-x) or exp(-x^2), over an interval such as [-1,+1], [0,+oo) or (-oo,+oo).

HERMITE_RULE, a FORTRAN90 program which generates a Gauss-Hermite quadrature rule on request.

HERMITE_TEST_INT, a FORTRAN90 library which defines test integrands for Hermite integrals with interval (-oo,+oo) and density exp(-x^2).

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_HERMITE, a FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Hermite 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.

Source Code:

List of Routines:

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


Last revised on 08 June 2013.