legendre_fast_rule, a FORTRAN90 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_fast_rule n a bwhere
The computer code and data files described and made available on this web page are distributed under the MIT license
legendre_fast_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
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quadrature_rules_legendre, a dataset directory which contains triples of files defining standard Gauss-Legendre quadrature rules.
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test_int, a FORTRAN90 code which contains number of functions that may be used as test integrands for quadrature rules in 1D.