GEN_LAGUERRE_RULE
Generalized Gauss-Laguerre Quadrature Rules

GEN_LAGUERRE_RULE is a FORTRAN90 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
      

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

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

        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
      

        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 order alpha option output

• 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.

Licensing:

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

Related Data and Programs:

CHEBYSHEV1_RULE, is a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, is a FORTRAN90 program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE is a FORTRAN90 program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, is a FORTRAN90 program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE is a FORTRAN90 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 FORTRAN90 program which computes a Gauss-Hermite quadrature rule.

INT_EXACTNESS, is a FORTRAN90 program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

INT_EXACTNESS_GEN_LAGUERRE is a FORTRAN90 program which checks the polynomial exactness of a generalized Gauss-Laguerre quadrature rule.

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

JACOBI_RULE is a FORTRAN90 program which computes a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE is a FORTRAN90 program which computes a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, is a FORTRAN90 program which computes a Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, is a FORTRAN90 program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

PATTERSON_RULE, is a FORTRAN90 program which computes a Gauss-Patterson quadrature rule.

PRODUCT_FACTOR is a FORTRAN90 program which constructs a product rule from distinct 1D factor rules.

PRODUCT_RULE is a FORTRAN90 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.

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

TEST_INT_LAGUERRE is a FORTRAN90 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,
   Gaussian Quadrature Formulas,
   Prentice Hall, 1966,
   LC: QA299.4G3S7.

Source Code:

• gen_laguerre_rule.f90, the source code.
• gen_laguerre_rule.csh, commands to compile the source code.

Examples and Tests:

• output_cpp.txt, printed output from the command

  
              gen_laguerre_rule 4 0.5 0 C++
            

• output_f77.txt, printed output from the command

  
              gen_laguerre_rule 4 -0.5 0 F77
            

• output_f90.txt, printed output from the command

  
              gen_laguerre_rule 4 0.0 0 F90
            

• output_f90_modified.txt, printed output from the command

  
              gen_laguerre_rule 4 0.0 1 F90
            

• output_mat.txt, printed output from the command

  
              gen_laguerre_rule 8 1.5 0 MAT
            

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

  
              gen_laguerre_rule 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 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 4 0.5 0 gen_lag_o4_a0.5
            

List of Routines:

• MAIN is the main program for GEN_LAGUERRE_RULE.
• 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.
• DTABLE_WRITE0 writes a DTABLE file with no headers.
• GEN_LAGUERRE_COMPUTE computes a generalized Gauss-Laguerre quadrature rule.
• GEN_LAGUERRE_HANDLE computes a generalized Gauss-Laguerre rule and outputs it.
• GEN_LAGUERRE_RECUR evaluates a generalized Laguerre polynomial.
• GEN_LAGUERRE_ROOT seeks roots of a generalized Laguerre polynomial.
• GET_UNIT returns a free FORTRAN unit number.
• R8_EPSILON returns the R8 roundoff unit.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HUGE returns a very large R8.
• S_BLANK_DELETE removes blanks from a string, left justifying the remainder.
• S_TO_I4 reads an I4 from a string.
• S_TO_R8 reads an R8 from a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

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