Order N Computation of Legendre Quadrature Rule

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].


legendre_rule_fast n a b


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 FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CLENSHAW_CURTIS_RULE, a C program which defines a Clenshaw Curtis quadrature rule.

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

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

LEGENDRE_EXACTNESS, a C program which tests the monomial exactness of quadrature rules for the Legendre problem of integrating a function with density 1 over the interval [-1,+1].

LEGENDRE_RULE, a C program which computes a 1D Gauss-Legendre quadrature rule.

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

QUADRULE, a C library which defines quadrature rules for 1D domains.


  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:

Examples and Tests:

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

List of Routines:

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

Last revised on 22 October 2009.