# GEN_LAGUERRE_RULE_SS Generalized Gauss-Laguerre Quadrature Rules

GEN_LAGUERRE_RULE_SS is a C++ program which generates a specific generalized Gauss-Laguerre quadrature rule, based on user input.

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.

Both the basic and generalized Gauss-Laguerre quadrature rules are 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.

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

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

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

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

        Integral ( 0 <= x < +oo ) x^alpha * 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 x^alpha * 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 generalized 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))


Note that when the real paramater alpha is equal to 0.0, we recover the basic (that is, "non-generalized") Gauss-Laguerre quadrature rule. Also note that the generalized Gauss-Laguerre rule is only defined for parameter values alpha which are greater than -1.0.

### Usage:

gen_laguerre_rule_ss order alpha option output
where
• order is the number of points in the quadrature rule. A typical value might be 4, 8, or 16.
• alpha is the parameter for the generalized Gauss-Laguerre quadrature rule. The value of alpha may be any real value greater than -1. Specifying alpha=0.0 results in the basic rule.
• option specifies the rule:
a 0 value requests the "standard" integration rule for
            Integral ( 0 <= x < +oo ) x^ALPHA 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.

### Languages:

GEN_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 computes a generalized Gauss-Hermite quadrature rule.

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

HERMITE_RULE is a C++ program which computes 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 computes a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE is a C++ program which computes a Gauss-Laguerre 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 Gauss-Laguerre 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 Gauss-Laguerre rules.

### Reference:

1. Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
2. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
3. 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.
4. Arthur Stroud, Don Secrest,
Prentice Hall, 1966,
LC: QA299.4G3S7.

### Examples and Tests:

• output_cpp.txt, printed output from the command

gen_laguerre_rule_ss 4 0.5 0 C++

• output_f77.txt, printed output from the command

gen_laguerre_rule_ss 4 -0.5 0 F77

• output_f90.txt, printed output from the command

gen_laguerre_rule_ss 4 0.0 0 F90

• output_f90_modified.txt, printed output from the command

gen_laguerre_rule_ss 4 0.0 1 F90

• output_mat.txt, printed output from the command

gen_laguerre_rule_ss 8 1.5 0 MAT

• gen_lag_o4_a0.5_r.txt, the region file created by the command

gen_laguerre_rule_ss 4 0.5 0 gen_lag_o4_a0.5

• gen_lag_o4_a0.5_w.txt, the weight file created by the command

gen_laguerre_rule_ss 4 0.5 0 gen_lag_o4_a0.5

• gen_lag_o4_a0.5_x.txt, the abscissa file created by the command

gen_laguerre_rule_ss 4 0.5 0 gen_lag_o4_a0.5


### List of Routines:

• MAIN is the main program for GEN_LAGUERRE_RULE_SS.
• GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre quadrature rule.
• GEN_LAGUERRE_RULE_HANDLE computes a generalized Gauss-Laguerre rule and outputs it.
• GEN_LAGUERRE_RECUR evaluates a generalized Laguerre polynomial.
• GEN_LAGUERRE_ROOT improves a root of a generalized 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 02 February 2010.