jburkardt

LEGENDRE_RULE_FAST\ Order N Computation of Legendre Quadrature Rule {#legendre_rule_fast-order-n-computation-of-legendre-quadrature-rule align=”center”} ===============================================


LEGENDRE_RULE_FAST is a C++ 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) dx

is 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].

Usage: {#usage align=”center”}

legendre_rule_fast n a b

where

Licensing: {#licensing align=”center”}

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages: {#languages align=”center”}

LEGENDRE_RULE_FAST is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

CHEBYSHEV1_RULE, a C++ program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, is a C++ program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE is a C++ program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, a C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a C++ program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, a C++ program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, a C++ program which can compute and print a Gauss-Hermite quadrature rule.

INT_EXACTNESS_LEGENDRE, a C++ program which checks the polynomial exactness of a Gauss-Legendre quadrature rule.

JACOBI_RULE, a C++ program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a C++ program which can compute and print a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, is a C++ program which can compute and print a Gauss-Legendre quadrature rule.

PATTERSON_RULE, is a C++ program which computes a Gauss-Patterson quadrature rule.

PRODUCT_RULE, a C++ program which constructs a product rule from 1D factor rules.

QUADRATURE_RULES_LEGENDRE, a dataset directory which contains triples of files defining standard Gauss-Legendre quadrature rules.

QUADRULE, a C++ library which defines 1-dimensional quadrature rules.

SANDIA_RULES, a C++ library which produces 1D quadrature rules of Chebyshev, Clenshaw Curtis, Fejer 2, Gegenbauer, generalized Hermite, generalized Laguerre, Hermite, Jacobi, Laguerre, Legendre and Patterson types.

TANH_SINH_RULE, a C++ program which computes and writes out a tanh-sinh quadrature rule of given order.

Reference: {#reference align=”center”}

  1. Andreas Glaser, Xiangtao Liu, Vladimir Rokhlin,\ A fast algorithm for the calculation of the roots of special functions,\ SIAM Journal on Scientific Computing,\ Volume 29, Number 4, pages 1420-1438, 2007.

Source Code: {#source-code align=”center”}

Examples and Tests: {#examples-and-tests align=”center”}

The following files were created by the command legendre_rule_fast 15 0.0 2.0:

List of Routines: {#list-of-routines align=”center”}

You can go up one level to the C++ source codes.


Last revised on 22 October 2009.