LEGENDRE_RULE_FAST is a FORTRAN90 program which implements a fast algorithm for the computation of the points and weights of the Gauss-Legendre quadrature rule.
The standard algorithm for computing the N points and weights of such a rule is by Golub and Welsch. It sets up and solves an eigenvalue problem, whose solution requires work of order N*N.
By contrast, the fast algorithm, by Glaser, Liu and Rokhlin, can compute the same information expending work of order N. For quadrature problems requiring high accuracy, where N might be 100 or more, the fast algorithm provides a significant improvement in speed.
The Gauss-Legendre quadrature rule is designed for the interval [-1,+1].
The Gauss-Legendre quadrature assumes that the integrand has the form:
Integral ( -1 <= x <= +1 ) f(x) dx
The standard Gauss-Legendre quadrature rule is used as follows:
Integral ( -1 <= x <= +1 ) f(x) dxis to be approximated by
Sum ( 1 <= i <= order ) w(i) * f(x(i))
This program allows the user to request that the rule be transformed from the standard interval [-1,+1] to the interval [a,b].
legendre_rule_fast n a bwhere
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
LEGENDRE_RULE_FAST is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
CHEBYSHEV1_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.
CHEBYSHEV2_RULE, a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.
CLENSHAW_CURTIS_RULE, a FORTRAN90 program which defines a Clenshaw Curtis quadrature rule.
GEGENBAUER_RULE, a FORTRAN90 program which can compute and print a Gauss-Gegenbauer quadrature rule.
GEN_HERMITE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Hermite quadrature rule.
GEN_LAGUERRE_RULE, a FORTRAN90 program which can compute and print a generalized Gauss-Laguerre quadrature rule.
HERMITE_RULE, a FORTRAN90 program which can compute and print a Gauss-Hermite quadrature rule.
INT_EXACTNESS_LEGENDRE, a FORTRAN90 program which checks the polynomial exactness of a Gauss-Legendre quadrature rule.
INTLIB, a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.
JACOBI_RULE, a FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.
LAGUERRE_RULE, a FORTRAN90 program which can compute and print a Gauss-Laguerre quadrature rule.
LEGENDRE_RULE, a FORTRAN90 program which can compute and print a Gauss-Legendre quadrature rule.
PATTERSON_RULE, a FORTRAN90 program which computes a Gauss-Patterson quadrature rule.
PRODUCT_RULE, a FORTRAN90 program which constructs a product rule from 1D factor rules.
QUADPACK, a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.
QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.
QUADRATURE_RULES_LEGENDRE, a dataset directory which contains triples of files defining standard Gauss-Legendre quadrature rules.
QUADRULE, a FORTRAN90 library which defines 1-dimensional quadrature rules.
TANH_SINH_RULE, a FORTRAN90 program which computes and writes out a tanh-sinh quadrature rule of given order.
TEST_INT, a FORTRAN90 library which contains number of functions that may be used as test integrands for quadrature rules in 1D.
The following files were created by the command legendre_rule_fast 15 -1 1:
You can go up one level to the FORTRAN90 source codes.