hermite_integrands, a FORTRAN90 code which defines integration problems over infinite intervals of the form (-oo,+oo).
The test integrands would normally be used to testing one dimensional quadrature software. It is possible to invoke a particular function by index, or to try out all available functions, as demonstrated in the sample calling program.
For a given integrand function f(x), the problem is to estimate
I(f) = integral ( -oo < x < +oo ) w(x) * f(x) dx
We consider three variations of the problem, depending on the form of the weight factor w(x):
Integral ( -oo < x < +oo ) f(x) dx
Integral ( -oo < x < +oo ) exp(-x*x) f(x) dx
Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx
For option 0, the test integrands have the form:
For option 1, the test integrands have the form:
For option 2, the test integrands have the form:
The library includes not just the integrand, but also the exact value of the integral (or, typically, an estimate of this value), and a title for the problem. Thus, for each integrand function, several routines are supplied. For instance, for function #1, we have the routines:
Moreover, since the same interface is used for each function, if you wish to work with problem 5 instead, you simply change the "01" to "05" in your routine calls.
If you wish to call all of the functions, then you simply use the generic interface, which requires you to specify the problem number as an extra input argument:
The computer code and data files described and made available on this web page are distributed under the MIT license
hermite_integrands is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
HERMITE_EXACTNESS, a FORTRAN90 code which tests the polynomial exactness of Gauss-Hermite quadrature rules.
HERMITE_RULE, a FORTRAN90 code which can compute and print a Gauss-Hermite quadrature rule.
LAGUERRE_TEST_INT, a FORTRAN90 code which defines test integrands for quadrature rules for estimating the integral of a function with density exp(-x) over the interval [0,+oo).
QUADRULE, a FORTRAN90 code which defines various 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, a FORTRAN90 code which defines some test integration problems over finite intervals.