hermite_rule, a FORTRAN90 code which computes and print a Gauss-Hermite quadrature rule for estimating the integral of a function with density exp(-x^2) over the interval (-oo,+oo).
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 ) dxis 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 n a b scale filenamewhere
The computer code and data files made available on this web page are distributed under the GNU LGPL license.
hermite_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
ALPERT_RULE, a FORTRAN90 code which can set up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.
CCN_RULE, a FORTRAN90 code which defines a nested Clenshaw Curtis quadrature rule.
CHEBYSHEV1_RULE, a FORTRAN90 code which can compute and print a Gauss-Chebyshev type 1 quadrature rule.
CHEBYSHEV2_RULE, a FORTRAN90 code which can compute and print a Gauss-Chebyshev type 2 quadrature rule.
CLENSHAW_CURTIS_RULE, a FORTRAN90 code which defines a Clenshaw Curtis quadrature rule.
GEGENBAUER_RULE, a FORTRAN90 code which can compute and print a Gauss-Gegenbauer quadrature rule.
GEN_HERMITE_RULE, a FORTRAN90 code which can compute and print a generalized Gauss-Hermite quadrature rule.
GEN_LAGUERRE_RULE, a FORTRAN90 code which can compute and print a generalized Gauss-Laguerre quadrature rule.
HERMITE_EXACTNESS, a FORTRAN90 code which tests the polynomial exactness of Gauss-Hermite quadrature rules.
HERMITE_POLYNOMIAL, a FORTRAN90 code which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.
JACOBI_RULE, a FORTRAN90 code which can compute and print a Gauss-Jacobi quadrature rule.
LAGUERRE_RULE, a FORTRAN90 code which can compute and print a Gauss-Laguerre quadrature rule.
LEGENDRE_RULE, a FORTRAN90 code which computes a Gauss-Legendre quadrature rule.
LEGENDRE_RULE_FAST, a FORTRAN90 code which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.
LOGNORMAL_RULE, a FORTRAN90 code which can compute and print a quadrature rule for functions of a variable whose logarithm is normally distributed.
PATTERSON_RULE, a FORTRAN90 code which returns the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.
PATTERSON_RULE_COMPUTE, a FORTRAN90 code which computes the points and weights of a 1D Gauss-Patterson quadrature rule of order 1, 3, 7, 15, 31, 63, 127, 255 or 511.
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.
QUADRULE, a FORTRAN90 code which contains 1-dimensional quadrature rules.
TRUNCATED_NORMAL_RULE, a FORTRAN90 code which computes a quadrature rule for a normal distribution that has been truncated to [A,+oo), (-oo,B] or [A,B].