LAGUERRE_RULE_SS
Gauss-Laguerre Quadrature Rules

LAGUERRE_RULE_SS is a C++ program which generates a specific Gauss-Laguerre quadrature rule, based on user input, using a Stroud and Secrest rule.

The rule can be output as text in a standard programming language, or the data can be written to three files for easy use as input to other programs.

The Gauss-Laguerre quadrature rule is designed to approximate integrals on semi-infinite intervals. By default, this interval is [0,oo). If this default is not suitable, it is possible to redefine the abscissas and weights for the interval [A,oo) instead.

The Gauss-Laguerre quadrature assumes that the integrand we are considering has a form like:

        Integral ( 0 <= x < +oo ) exp(-x) f(x) dx

where the factor exp(-x) is regarded as a weight factor.

The standard Gauss-Laguerre quadrature rule is used as follows:

        Integral ( 0 <= x < +oo ) exp(-x) f(x) dx

is to be approximated by

        Sum ( 1 <= i <= order ) w(i) * f(x(i))

Although the standard rule is defined in terms of the product of the integral weight function exp(-x) and a function f(x), there may be cases when it is more convenient to think that we are simply approximating

        Integral ( 0 <= x < +oo ) f(x) dx

The standard rule can easily be modified, by adjusting the weights, so that the computation can be done in this form. The program allows the user to specify, through the parameter OPTION, whether the standard rule is to be computed (OPTION=0), or the modified rule (OPTION=1).

The modified Gauss-Laguerre quadrature rule is used as follows:

        Integral ( 0 <= x < +oo ) f(x) dx

is to be approximated by

        Sum ( 1 <= i <= order ) w(i) * f(x(i))

Usage:

laguerre_rule_ss order option output

where

order is the number of points in the quadrature rule. A typical value might be 4, 8, or 16.

option choose the rule:
a 0 value requests the "standard" integration rule for

            Integral ( 0 <= x < +oo ) exp(-x) f(x) dx

a 1 value requests the "modified" integration rule for

            Integral ( 0 <= x < +oo )         f(x) dx

output specifies how the rule is to be reported:
- C++, print as C++ text;
- F77, print as FORTRAN77 text;
- F90, print as FORTRAN90 text;
- MAT, print as MATLAB text;
- file, written to three files, file_w.txt, file_x.txt, and file_r.txt, containing the weights, abscissas, and interval limits.

Licensing:

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

Languages:

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

Related Data and Programs:

CHEBYSHEV1_RULE, is 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, is a C++ program which can compute and print a Gauss-Gegenbauer quadrature rule.

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

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

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

INT_EXACTNESS, is a C++ program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

INT_EXACTNESS_LAGUERRE, is a C++ program which checks the polynomial exactness of a Gauss-Laguerre quadrature rule.

INTLIB is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

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

LEGENDRE_RULE, is a C++ program which computes a Gauss-Legendre quadrature rule.

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

PRODUCT_FACTOR is a C++ program which constructs a product rule from distinct 1D factor rules.

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

QUADPACK is a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

QUADRATURE_RULES is a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_LAGUERRE is a dataset directory which contains triples of files defining standard Laguerre quadrature rules.

QUADRULE is a C++ library which contains 1-dimensional quadrature rules.

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

TEST_INT is a FORTRAN90 library which defines functions that may be used as test integrands for quadrature rules in 1D.

TEST_INT_LAGUERRE is a C++ library which defines test integrands for integration over [A,+oo).

Reference:

Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
Philip Rabinowitz, George Weiss,
Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form $\int_0^{\infty} exp(-x) x^n f(x) dx$,
Mathematical Tables and Other Aids to Computation,
Volume 13, Number 68, October 1959, pages 285-294.
Arthur Stroud, Don Secrest,
Gaussian Quadrature Formulas,
Prentice Hall, 1966,
LC: QA299.4G3S7.

Source Code:

laguerre_rule_ss.cpp, the source code.

Examples and Tests:

output_cpp.txt, printed output from the command


            laguerre_rule_ss 4 0 C++

output_f77.txt, printed output from the command


            laguerre_rule_ss 4 0 F77

output_f90.txt, printed output from the command


            laguerre_rule_ss 4 0 F90

output_f90_modified.txt, printed output from the command


            laguerre_rule_ss 4 1 F90

output_mat.txt, printed output from the command


            laguerre_rule_ss 8 0 MAT

lag_o4_r.txt, the region file created by the command


            laguerre_rule_ss 4 0 lag_o4

lag_o4_w.txt, the weight file created by the command


            laguerre_rule_ss 4 0 lag_o4

lag_o4_x.txt, the abscissa file created by the command


            laguerre_rule_ss 4 0 lag_o4

lag_o4_modified_r.txt, the region file created by the command


            laguerre_rule_ss 4 1 lag_o4

lag_o4_modified_w.txt, the weight file created by the command


            laguerre_rule_ss 4 1 lag_o4

lag_o4_modified_x.txt, the abscissa file created by the command


            laguerre_rule_ss 4 1 lag_o4

List of Routines:

MAIN is the main program for LAGUERRE_RULE_SS.
LAGUERRE_COMPUTE computes a Gauss-Laguerre quadrature rule.
LAGUERRE_HANDLE computes the requested Gauss-Laguerre rule and outputs it.
LAGUERRE_RECUR evaluates a Laguerre polynomial.
LAGUERRE_ROOT improves a root of a Laguerre polynomial.
R8_ABS returns the absolute value of an R8.
R8_EPSILON returns the R8 roundoff unit.
R8_GAMMA evaluates Gamma(X) for a real argument.
R8_HUGE returns a "huge" R8.
R8MAT_WRITE writes an R8MAT file with no header.
TIMESTAMP prints the current YMDHMS date as a time stamp.

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

Last revised on 03 February 2010.

LAGUERRE_RULE_SS Gauss-Laguerre Quadrature Rules