gen_hermite_exactness, a Python code which investigates the polynomial exactness of a generalized Gauss-Hermite quadrature rule for the infinite interval (-oo,+oo).
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) dxwhere 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) dxmay 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) dxand 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) dxmay 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
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:
gen_hermite_exactness ( 'prefix', degree_max, alpha, option )where
If the arguments are not supplied on the command line, the program will prompt for them.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
gen_hermite_exactness is available in a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.
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GEN_HERM_O8_A1.0 is a standard generalized Gauss-Hermite order 8 rule with ALPHA = 1.0.
gen_hermite_exactness ( 'gen_herm_o8_a1.0', 18, 1.0, 0 )
GEN_HERM_O8_A1.0_MODIFIED is a modified generalized Gauss-Hermite order 8 rule with ALPHA = 1.0.
gen_hermite_exactness ( 'gen_herm_o8_a1.0_modified', 18, 1.0, 1 )