legendre_rule_fast, a MATLAB code 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, b )where
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 MATLAB code which can compute and print a gauss-chebyshev type 1 quadrature rule.
chebyshev2_rule, a MATLAB code which can compute and print a gauss-chebyshev type 2 quadrature rule.
clenshaw_curtis_rule, a MATLAB code which defines a clenshaw curtis quadrature rule.
gegenbauer_rule, a MATLAB code which can compute and print a gauss-gegenbauer quadrature rule.
gen_hermite_rule, a MATLAB code which can compute and print a generalized gauss-hermite quadrature rule.
gen_laguerre_rule, a MATLAB code which can compute and print a generalized gauss-laguerre quadrature rule.
hermite_rule, a MATLAB code which can compute and print a gauss-hermite quadrature rule.
jacobi_rule, a MATLAB code which can compute and print a gauss-jacobi quadrature rule.
laguerre_rule, a MATLAB code which can compute and print a gauss-laguerre quadrature rule.
legendre_rule, a MATLAB code which can compute and print a gauss-legendre quadrature rule.
patterson_rule, a MATLAB code which computes a gauss-patterson quadrature rule.
product_rule, a MATLAB code which constructs a product rule from 1d factor rules.