LAGUERRE_RULE
Gauss-Laguerre Quadrature Rules

LAGUERRE_RULE is an interactive FORTRAN90 program, using double precision arithmetic, which can generate a specific 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.

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
      

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

        Integral ( 0 <= x < +oo ) 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 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
      

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

Usage:

laguerre_rule order option output

order

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

option

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.

Related Data and Programs:

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

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

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

GEN_HERMITE_RULE, is an executable FORTRAN90 program which can compute and print a generalized Gauss-Hermite quadrature rule.

GEN_LAGUERRE_RULE, is an executable FORTRAN90 program which can compute and print a generalized Gauss-Laguerre quadrature rule.

HERMITE_RULE, is an executable FORTRAN90 program which can compute and print a Gauss-Hermite quadrature rule.

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

INT_EXACTNESS_LAGUERRE, is an executable FORTRAN90 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 an executable FORTRAN90 program which can compute and print a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE is also available in a C++ version and a MATLAB version.

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

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

PRODUCT_RULE is an executable 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 of triples of files defining standard Laguerre quadrature rules.

QUADRULE is a FORTRAN90 library which contains 1-dimensional quadrature rules.

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 integration over [A,+oo).

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:

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

Examples and Tests:

• output_cpp.txt, printed output from the command

  
              laguerre_rule 4 0 C++
            

• output_f77.txt, printed output from the command

  
              laguerre_rule 4 0 F77
            

• output_f90.txt, printed output from the command

  
              laguerre_rule 4 0 F90
            

• output_f90_modified.txt, printed output from the command

  
              laguerre_rule 4 1 F90
            

• output_mat.txt, printed output from the command

  
              laguerre_rule 8 0 MAT
            

• lag_o4_r.txt, the region file created by the command

  
              laguerre_rule 4 0 lag_o4
            

• lag_o4_w.txt, the weight file created by the command

  
              laguerre_rule 4 0 lag_o4
            

• lag_o4_x.txt, the abscissa file created by the command

  
              laguerre_rule 4 0 lag_o4
            

• lag_o4_modified_r.txt, the region file created by the command

  
              laguerre_rule 4 1 lag_o4
            

• lag_o4_modified_w.txt, the weight file created by the command

  
              laguerre_rule 4 1 lag_o4
            

• lag_o4_modified_x.txt, the abscissa file created by the command

  
              laguerre_rule 4 1 lag_o4
            

List of Routines:

• MAIN is the main program for 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_CLOSE_WRITE closes a file used to write a DTABLE.
• DTABLE_DATA_WRITE writes DTABLE data to a file.
• DTABLE_HEADER_WRITE writes the header to a DTABLE file.
• DTABLE_OPEN_WRITE opens a file to write a DTABLE.
• DTABLE_WRITE writes a DTABLE to a file.
• GET_UNIT returns a free FORTRAN unit number.
• LAGUERRE_COMPUTE computes a Gauss-Laguerre quadrature rule.
• LAGUERRE_HANDLE tests computes the requested rule and outputs it.
• LAGUERRE_RECUR finds the value and derivative of a Laguerre polynomial.
• LAGUERRE_ROOT improves an approximate root of a Laguerre polynomial.
• R8_GAMMA computes the gamma function.
• R8_HUGE returns the largest legal 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.
• TIMESTRING writes the current YMDHMS date into a string.

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