LEGENDRE_RULE_FAST
Gauss-Legendre Quadrature Rules

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) dx
      

        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:

legendre_rule_fast n a b

• n is the order (number of points);
• a is the left endpoint (often -1.0 or 0.0);
• b is the right endpoint (usually 1.0).

Licensing:

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

Languages:

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

Related Data and Programs:

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.

Reference:

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:

• legendre_rule_fast.f90, the source code.

Examples and Tests:

The following files were created by the command legendre_rule_fast 15 -1 1:

• leg_o15_r.txt, the region file.
• leg_o15_w.txt, the weight file.
• leg_o15_x.txt, the abscissa file.

List of Routines:

• MAIN is the main program for LEGENDRE_RULE_FAST.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DIGIT_TO_CH returns the character representation of a decimal digit.
• GET_UNIT returns a free FORTRAN unit number.
• I4_TO_S_LEFT converts an I4 to a left-justified string.
• LEGENDRE_COMPUTE_GLR: Legendre quadrature by the Glaser-Liu-Rokhlin method.
• LEGENDRE_COMPUTE_GLR0 gets a starting value for the fast algorithm.
• LEGENDRE_COMPUTE_GLR1 gets the complete set of Legendre points and weights.
• LEGENDRE_COMPUTE_GLR2 finds the first real root.
• LEGENDRE_HANDLE computes the requested Gauss-Legendre rule and outputs it.
• R8MAT_WRITE writes an R8MAT file.
• RESCALE rescales a Legendre quadrature rule from [-1,+1] to [A,B].
• RK2_LEG advances the value of X(T) using a Runge-Kutta method.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 value from a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TS_MULT evaluates a polynomial.

You can go up one level to the FORTRAN90 source codes.