hermite_rule, a C++ code which generates a specific Gauss-Hermite quadrature rule, based on user input.
The rule is written to three files for easy use as input to other programs.
The Gauss-Hermite quadrature rule is used as follows:
c * Integral ( -oo < x < +oo ) f(x) exp ( - b * ( x - a )^2 ) dx
is to be approximated by
Sum ( 1 <= i <= order ) w(i) * f(x(i))
Generally, a Gauss-Hermite quadrature rule of n points will
produce the exact integral when f(x) is a polynomial of degree
2n-1 or less.
The value of C in front of the integral depends on the user's choice of the SCALE parameter:
hermite_rule order a b scale filenamewhere
The computer code and data files described and made available on this web page are distributed under the MIT license
hermite_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
CCN_RULE, a C++ code which defines a nested Clenshaw Curtis quadrature rule.
CHEBYSHEV_POLYNOMIAL, a C++ code which evaluates the Chebyshev polynomial and associated functions.
CHEBYSHEV1_RULE, a C++ code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.
CHEBYSHEV2_RULE, a C++ code which can compute and print a Gauss-Chebyshev type 2 quadrature rule.
CLENSHAW_CURTIS_RULE, a C++ code which defines a Clenshaw Curtis quadrature rule.
GEGENBAUER_RULE, a C++ code which can compute and print a Gauss-Gegenbauer quadrature rule.
GEN_HERMITE_RULE, a C++ code which can compute and print a generalized Gauss-Hermite quadrature rule.
GEN_LAGUERRE_RULE, a C++ code which can compute and print a generalized Gauss-Laguerre quadrature rule.
HERMITE_EXACTNESS, a C++ code which tests the polynomial exactness of Gauss-Hermite quadrature rules for estimating the integral of a function with density exp(-x^2) over the interval (-oo,+oo).
JACOBI_RULE, a C++ code which can compute and print a Gauss-Jacobi quadrature rule.
LAGUERRE_RULE, a C++ code which can compute and print a Gauss-Laguerre quadrature rule.
LATTICE_RULE, a C++ code which approximates M-dimensional integrals using lattice rules.
LEGENDRE_RULE, a C++ code which computes a Gauss-Legendre quadrature rule.
LINE_FELIPPA_RULE, a C++ code which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.
PATTERSON_RULE, a C++ code which computes a Gauss-Patterson quadrature rule.
QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.
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.
TRUNCATED_NORMAL_RULE, a C++ code which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian distribution, that has been truncated to [A,+oo), (-oo,B] or [A,B].