INT_EXACTNESS_LAGUERRE
Exactness of Gauss-Laguerre Quadrature Rules

INT_EXACTNESS_LAGUERRE is a FORTRAN90 program which investigates the polynomial exactness of a Gauss-Laguerre quadrature rule for the semi-infinite interval [0,+oo) or [A,+oo).

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

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

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

A standard Gauss-Laguerre quadrature rule is a set of n positive weights w and abscissas x so that

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

may be approximated by

        Sum ( 1 <= I <= N ) w(i) * f(x(i))

It is often convenient to consider approximating integrals in which the weighting factor exp(-x) is implicit. In that case, we are looking at approximating

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

and it is easy to modify a standard Gauss-Laguerre quadrature rule to handle this case directly.

A modified Gauss-Laguerre quadrature rule is a set of n positive weights w and abscissas x so that

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

may be approximated by

        Sum ( 1 <= I <= N ) w(i) * f(x(i))

When using a Gauss-Laguerre quadrature rule, it's important to know whether the rule has been developed for the standard or modified cases. Basically, the only change is that the weights of the modified rule have been multiplied by an exponential factor evaluated at the corresponding abscissa.

For a standard Gauss-Laguerre rule, polynomial exactness is defined in terms of the function f(x). That is, we say the rule is exact for polynomials up to degree DEGREE_MAX if, for any polynomial f(x) of that degree or less, the quadrature rule will produce the exact value of

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

For a modified Gauss-Laguerre rule, polynomial exactness is defined in terms of the function f(x) divided by the implicit weight function. That is, we say a modified Gauss-Laguerre rule is exact for polynomials up to degree DEGREE_MAX if, for any integrand f(x) with the property that exp(+x) * f(x) is a polynomial of degree no more than DEGREE_MAX, the quadrature rule will product the exact value of:

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

The program starts at DEGREE = 0, and then proceeds to DEGREE = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of DEGREE, the program generates the corresponding monomial term, applies the quadrature rule to it, and determines the quadrature error. The program uses a scaling factor on each monomial so that the exact integral should always be 1; therefore, each reported error can be compared on a fixed scale.

If the program understands that the rule being considered is a modified rule, then the monomials are multiplied by exp(-x) when performing the exactness test.

Since

        Integral ( 0 <= x < +oo ) exp(-x) xⁿ dx = n!

our test monomial functions, in order to integrate to 1, will be normalized to:

        Integral ( 0 <= x < +oo ) exp(-x) xⁿ / n! dx

It should be clear that accuracy will be rapidly lost as n increases.

The program is very flexible and interactive. The quadrature rule is defined by three files, to be read at input, and the maximum degree top be checked is specified by the user as well.

Note that the three files that define the quadrature rule are assumed to have related names, of the form

prefix_x.txt
prefix_w.txt
prefix_r.txt

When running the program, the user only enters the common prefix part of the file names, which is enough information for the program to find all three files.

For information on the form of these files, see the QUADRATURE_RULES directory listed below.

The exactness results are written to an output file with the corresponding name:

prefix_exact.txt

Usage:

int_exactness_laguerre prefix degree_max option

where

prefix is the common prefix for the files containing the abscissa, weight and region information of the quadrature rule;
degree_max is the maximum monomial degree to check. This would normally be a relatively small nonnegative number, such as 5, 10 or 15.
option is:
0: a standard rule for integrating exp(-x)*f(x).
1: a modified rule for integrating f(x).

If the arguments are not supplied on the command line, the program will prompt for them.

Licensing:

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

Languages:

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

Related Data and Programs:

HERMITE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

INT_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of a quadrature rule for a finite interval.

INT_EXACTNESS_CHEBYSHEV1, a FORTRAN90 program which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.

INT_EXACTNESS_CHEBYSHEV2, a FORTRAN90 program which tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.

INT_EXACTNESS_GEGENBAUER, a FORTRAN90 program which tests the polynomial exactness of Gauss-Gegenbauer quadrature rules.

INT_EXACTNESS_GEN_HERMITE, a FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.

INT_EXACTNESS_GEN_LAGUERRE, a FORTRAN90 program which tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.

INT_EXACTNESS_JACOBI, a FORTRAN90 program which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

INT_EXACTNESS_LEGENDRE, a FORTRAN90 program which tests the polynomial exactness of Gauss-Legendre quadrature rules.

INTEGRAL_TEST, a FORTRAN90 program which uses test integrals to measure the effectiveness of certain sets of quadrature rules.

INTLIB, a FORTRAN90 library which numerically estimate integrals in one dimension.

LAGUERRE_RULE, a FORTRAN90 program which generates a Gauss-Laguerre quadrature rule.

NINT_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of multidimensional integration rules.

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

QUADRATURE_RULES_LAGUERRE, a dataset directory which contains sets of files that define Gauss-Laguerre quadrature rules.

QUADRULE, a FORTRAN90 library which defines quadrature rules on a variety of intervals with different weight functions.

STROUD, a FORTRAN90 library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and multiple dimensions.

TEST_INT_LAGUERRE, a FORTRAN90 library which defines integrand functions that can be approximately integrated by a Gauss-Laguerre rule.

Reference:

Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.

Source Code:

int_exactness_laguerre.f90, the source code.
int_exactness_laguerre.sh, commands to compile the source code.

Examples and Tests:

LAG_O1 is a standard Gauss-Laguerre order 1 rule.

lag_o1_x.txt, the abscissas of the rule.
lag_o1_w.txt, the weights of the rule.
lag_o1_r.txt, defines the region for the rule.
lag_o1_exact.txt, the results of the command
int_exactness_laguerre lag_o1 5 0

LAG_O2 is a standard Gauss-Laguerre order 2 rule.

lag_o2_x.txt, the abscissas of the rule.
lag_o2_w.txt, the weights of the rule.
lag_o2_r.txt, defines the region for the rule.
lag_o2_exact.txt, the results of the command
int_exactness_laguerre lag_o2 5 0

LAG_O4 is a standard Gauss-Laguerre order 4 rule.

lag_o4_x.txt, the abscissas of the rule.
lag_o4_w.txt, the weights of the rule.
lag_o4_r.txt, defines the region for the rule.
lag_o4_exact.txt, the results of the command
int_exactness_laguerre lag_o4 10 0

LAG_O8 is a standard Gauss-Laguerre order 8 rule.

lag_o8_x.txt, the abscissas of the rule.
lag_o8_w.txt, the weights of the rule.
lag_o8_r.txt, defines the region for the rule.
lag_o8_exact.txt, the results of the command
int_exactness_laguerre lag_o8 18 0

LAG_O16 is a standard Gauss-Laguerre order 16 rule.

lag_o16_x.txt, the abscissas of the rule.
lag_o16_w.txt, the weights of the rule.
lag_o16_r.txt, defines the region for the rule.
lag_o16_exact.txt, the results of the command
int_exactness_laguerre lag_o16 35 0

LAG_O1_MODIFIED is a modified Gauss-Laguerre order 1 rule.

lag_o1_modified_x.txt, the abscissas of the rule.
lag_o1_modified_w.txt, the weights of the rule.
lag_o1_modified_r.txt, defines the region for the rule.
lag_o1_modified_exact.txt, the results of the command
int_exactness_laguerre lag_o1_modified 5 1

LAG_O2_MODIFIED is a modified Gauss-Laguerre order 2 rule.

lag_o2_modified_x.txt, the abscissas of the rule.
lag_o2_modified_w.txt, the weights of the rule.
lag_o2_modified_r.txt, defines the region for the rule.
lag_o2_modified_exact.txt, the results of the command
int_exactness_laguerre lag_o2_modified 5 1

LAG_O4_MODIFIED is a modified Gauss-Laguerre order 4 rule.

lag_o4_modified_x.txt, the abscissas of the rule.
lag_o4_modified_w.txt, the weights of the rule.
lag_o4_modified_r.txt, defines the region for the rule.
lag_o4_modified_exact.txt, the results of the command
int_exactness_laguerre lag_o4_modified 10 1

LAG_O8_MODIFIED is a modified Gauss-Laguerre order 8 rule.

lag_o8_modified_x.txt, the abscissas of the rule.
lag_o8_modified_w.txt, the weights of the rule.
lag_o8_modified_r.txt, defines the region for the rule.
lag_o8_modified_exact.txt, the results of the command
int_exactness_laguerre lag_o8_modified 18 1

LAG_O16_MODIFIED is a modified Gauss-Laguerre order 16 rule.

lag_o16_modified_x.txt, the abscissas of the rule.
lag_o16_modified_w.txt, the weights of the rule.
lag_o16_modified_r.txt, defines the region for the rule.
lag_o16_modified_exact.txt, the results of the command
int_exactness_laguerre lag_o16_modified 35 1

List of Routines:

MAIN is the main program for INT_EXACTNESS_LAGUERRE.
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.
FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
FILE_ROW_COUNT counts the number of row records in a file.
GET_UNIT returns a free FORTRAN unit number.
LAGUERRE_INTEGRAL evaluates a monomial Laguerre integral.
MONOMIAL_QUADRATURE_LAGUERRE applies a quadrature rule to a monomial.
R8_FACTORIAL computes the factorial of N, also denoted "N!".
R8MAT_DATA_READ reads data from an R8MAT file.
R8MAT_HEADER_READ reads the header from an R8MAT file.
S_TO_I4 reads an I4 from a string.
S_TO_R8 reads an R8 from a string.
S_TO_R8VEC reads an R8VEC from a string.
S_WORD_COUNT counts the number of "words" in a string.
TIMESTAMP prints the current YMDHMS date as a time stamp.

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

Last revised on 27 January 2010.

INT_EXACTNESS_LAGUERRE Exactness of Gauss-Laguerre Quadrature Rules