HERMITE_RULE_SS is a C++ program which generates a specific Gauss-Hermite quadrature rule, based on user input, using a Stroud and Secrest algorithm.
The rule can be output as text in a standard programming language, or the data can be written to three files for easy use as input to other programs.
The Gauss-Hermite quadrature rule is designed to approximate integrals on infinite intervals.
The Gauss Hermite quadrature assumes that the integrand we are considering has a form like:
Integral ( -oo < x < +oo ) w(x) * f(x) dxwhere the factor w(x) is regarded as a weight factor.
We consider three variations of the rule, 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
The corresponding Gauss-Hermite rule that uses order points will approximate the integral by
sum ( 1 <= i <= order ) w(i) * f(x(i))where, confusingly, w(i) is a vector of quadrature weights, which has no connection with the w(x) weight function.
hermite_rule_ss order option outputwhere
Integral ( -oo < x < +oo ) f(x) dxa 1 value requests a rule for the physicist weighted integral:
Integral ( -oo < x < +oo ) exp(-x*x) f(x) dxa 2 value requests a rule for the probabilist weighted integral:
Integral ( -oo < x < +oo ) exp(-x*x/2) f(x) dx
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
HERMITE_RULE_SS is available in a C++ version and a FORTRAN90 version and a MATLAB version.
CHEBYSHEV1_RULE, is a C++ program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.
CHEBYSHEV2_RULE, is a C++ program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.
CLENSHAW_CURTIS_RULE is a C++ program which defines a Clenshaw Curtis quadrature rule.
GEGENBAUER_RULE, is a C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.
GEN_HERMITE_RULE, is a C++ program which can compute and print a generalized Gauss-Hermite quadrature rule.
GEN_LAGUERRE_RULE, is a C++ program which can compute and print a generalized Gauss-Laguerre quadrature rule.
HERMITE_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.
INT_EXACTNESS, is a C++ program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.
INT_EXACTNESS_HERMITE, is a C++ program which checks the polynomial exactness of a Gauss-Hermite quadrature rule.
INTLIB is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.
JACOBI_RULE, is a C++ program which can compute and print a Gauss-Jacobi quadrature rule.
LAGUERRE_RULE, is a C++ program which can compute and print a Gauss-Laguerre quadrature rule.
LEGENDRE_RULE, is a C++ program which computes a Gauss-Legendre quadrature rule.
PATTERSON_RULE, is a C++ program which computes a Gauss-Patterson quadrature rule.
PRODUCT_FACTOR is an C++ program which constructs a product rule from distinct 1D factor rules.
PRODUCT_RULE is a C++ program which constructs a product rule from identical 1D factor rules.
QUADPACK is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.
QUADRATURE_RULES is a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.
QUADRATURE_RULES_HERMITE is a dataset directory of triples of files defining standard Hermite quadrature rules.
QUADRULE is a C++ library which contains 1-dimensional quadrature rules.
TANH_SINH_RULE, a C++ program which computes and writes out a tanh-sinh quadrature rule of given order.
TEST_INT is a FORTRAN90 library which defines functions that may be used as test integrands for quadrature rules in 1D.
TEST_INT_HERMITE is a C++ library which defines test integrands for integration over (-oo,+oo).
hermite_rule_ss 4 1 C++
hermite_rule_ss 4 1 F77
hermite_rule_ss 4 0 F90
hermite_rule_ss 4 1 F90
hermite_rule_ss 4 2 F90
hermite_rule_ss 8 1 MAT
hermite_rule_ss 4 0 herm_o4
hermite_rule_ss 4 0 herm_o4
hermite_rule_ss 4 0 herm_o4
hermite_rule_ss 4 1 herm_o4
hermite_rule_ss 4 1 herm_o4
hermite_rule_ss 4 1 herm_o4
hermite_rule_ss 4 2 herm_o4
hermite_rule_ss 4 2 herm_o4
hermite_rule_ss 4 2 herm_o4
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