INT_EXACTNESS_GEN_LAGUERRE
Exactness of Generalized Gauss-Laguerre Quadrature Rules

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

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

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

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

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

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

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

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

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

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

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

When using a generalized 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 divided by the weighting function evaluated at the corresponding abscissa.

For a standard generalized 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 ) x^alpha * exp(-x) * f(x) dx
      

For a modified generalized Gauss-Laguerre rule, polynomial exactness is defined in terms of the function f(x) divided by the implicit weighting function. That is, we say a modified generalized Gauss-Laguerre rule is exact for polynomials up to degree DEGREE_MAX if, for any integrand f(x) with the property that f(x)/(x^alpha*exp(-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 x^alpha * exp(-x) when performing the exactness test.

Since

        Integral ( 0 <= x < +oo ) x^alpha * exp(-x) * xn dx = gamma(n+alpha+1)
      

        Integral ( 0 <= x < +oo ) x^alpha * exp(-x) xn / gamma(n+alpha+1) dx
      

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

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_gen_laguerre ( 'prefix', degree_max, alpha, option )

• 'prefix' is a quoted string, 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.
• alpha is the value of the parameter, which should be a real number greater than -1. Setting alpha to 0.0 results in the basic (non-generalized) Gauss-Laguerre rule.
• option:
  0 indicates a standard rule for integrating x^alpha*exp(-x)*f(x).
  1 indicates 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_GEN_LAGUERRE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

GEN_LAGUERRE_RULE, a MATLAB program which can generate a generalized Gauss-Laguerre quadrature rule on request.

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

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

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

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

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

INT_EXACTNESS_HERMITE, a MATLAB program which tests the polynomial exactness of Gauss-Hermite quadrature rules.

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

INT_EXACTNESS_LAGUERRE, a MATLAB program which tests the polynomial exactness of Gauss-Laguerre quadrature rules.

INT_EXACTNESS_LEGENDRE, a MATLAB 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 estimates integrals in one dimension.

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

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

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

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

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

Reference:

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

Source Code:

• int_exactness_gen_laguerre.m, is the source code.

Examples and Tests:

GEN_LAG_O1_A0.5 is a standard generalized Gauss-Laguerre order 1 rule with ALPHA = 0.5.

• gen_lag_o1_a0.5_x.txt, the abscissas of the rule.
• gen_lag_o1_a0.5_w.txt, the weights of the rule.
• gen_lag_o1_a0.5_r.txt, defines the region for the rule.
• gen_lag_o1_a0.5_exact.txt, the results of the command

  int_exactness_gen_laguerre ( 'gen_lag_o1_a0.5', 5, 0.5, 0 )

GEN_LAG_O2_A0.5 is a standard generalized Gauss-Laguerre order 2 rule with ALPHA = 0.5.

• gen_lag_o2_a0.5_x.txt, the abscissas of the rule.
• gen_lag_o2_a0.5_w.txt, the weights of the rule.
• gen_lag_o2_a0.5_r.txt, defines the region for the rule.
• gen_lag_o2_a0.5_exact.txt, the results of the command

  int_exactness_gen_laguerre ( 'gen_lag_o2_a0.5', 5, 0.5, 0 )

GEN_LAG_O4_A0.5 is a standard generalized Gauss-Laguerre order 4 rule with ALPHA = 0.5.

• gen_lag_o4_a0.5_x.txt, the abscissas of the rule.
• gen_lag_o4_a0.5_w.txt, the weights of the rule.
• gen_lag_o4_a0.5_r.txt, defines the region for the rule.
• gen_lag_o4_a0.5_exact.txt, the results of the command

  int_exactness_gen_laguerre ( 'gen_lag_o4_a0.5', 10, 0.5, 0 )

GEN_LAG_O8_A0.5 is a standard generalized Gauss-Laguerre order 8 rule with ALPHA = 0.5.

• gen_lag_o8_a0.5_x.txt, the abscissas of the rule.
• gen_lag_o8_a0.5_w.txt, the weights of the rule.
• gen_lag_o8_a0.5_r.txt, defines the region for the rule.
• gen_lag_o8_a0.5_exact.txt, the results of the command

  int_exactness_gen_laguerre ( 'gen_lag_o8_a0.5', 18, 0.5, 0 )

GEN_LAG_O16_A0.5 is a standard generalized Gauss-Laguerre order 16 rule with ALPHA = 0.5.

• gen_lag_o16_a0.5_x.txt, the abscissas of the rule.
• gen_lag_o16_a0.5_w.txt, the weights of the rule.
• gen_lag_o16_a0.5_r.txt, defines the region for the rule.
• gen_lag_o16_a0.5_exact.txt, the results of the command

  int_exactness_gen_laguerre ( 'gen_lag_o16_a0.5', 35, 0.5, 0 )

GEN_LAG_O1_A0.5_MODIFIED is a modified generalized Gauss-Laguerre order 1 rule with ALPHA = 0.5.

• gen_lag_o1_a0.5_modified_x.txt, the abscissas of the rule.
• gen_lag_o1_a0.5_modified_w.txt, the weights of the rule.
• gen_lag_o1_a0.5_modified_r.txt, defines the region for the rule.
• gen_lag_o1_a0.5_modified_exact.txt, the results of the command

  int_exactness_gen_laguerre ( 'gen_lag_o1_a0.5_modified', 5, 0.5, 1 )

GEN_LAG_O2_A0.5_MODIFIED is a modified generalized Gauss-Laguerre order 2 rule with ALPHA = 0.5.

• gen_lag_o2_a0.5_modified_x.txt, the abscissas of the rule.
• gen_lag_o2_a0.5_modified_w.txt, the weights of the rule.
• gen_lag_o2_a0.5_modified_r.txt, defines the region for the rule.
• gen_lag_o2_a0.5_modified_exact.txt, the results of the command

  int_exactness_gen_laguerre ( 'gen_lag_o2_a0.5_modified', 5, 0.5, 1 )

GEN_LAG_O4_A0.5_MODIFIED is a modified generalized Gauss-Laguerre order 4 rule with ALPHA = 0.5.

• gen_lag_o4_a0.5_modified_x.txt, the abscissas of the rule.
• gen_lag_o4_a0.5_modified_w.txt, the weights of the rule.
• gen_lag_o4_a0.5_modified_r.txt, defines the region for the rule.
• gen_lag_o4_a0.5_modified_exact.txt, the results of the command

  int_exactness_gen_laguerre ( 'gen_lag_o4_a0.5_modified', 10, 0.5, 1 )

GEN_LAG_O8_A0.5_MODIFIED is a modified generalized Gauss-Laguerre order 8 rule with ALPHA = 0.5.

• gen_lag_o8_a0.5_modified_x.txt, the abscissas of the rule.
• gen_lag_o8_a0.5_modified_w.txt, the weights of the rule.
• gen_lag_o8_a0.5_modified_r.txt, defines the region for the rule.
• gen_lag_o8_a0.5_modified_exact.txt, the results of the command

  int_exactness_gen_laguerre ( 'gen_lag_o8_a0.5_modified', 18, 0.5, 1 )

GEN_LAG_O16_A0.5_MODIFIED is a modified generalized Gauss-Laguerre order 16 rule with ALPHA = 0.5.

• gen_lag_o16_a0.5_modified_x.txt, the abscissas of the rule.
• gen_lag_o16_a0.5_modified_w.txt, the weights of the rule.
• gen_lag_o16_a0.5_modified_r.txt, defines the region for the rule.
• gen_lag_o16_a0.5_modified_exact.txt, the results of the command

  int_exactness_gen_laguerre ( 'gen_lag_o16_a0.5_modified', 35, 0.5, 1 )

List of Routines:

• MAIN is the main program for INT_EXACTNESS_GEN_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.
• DTABLE_DATA_READ reads data from a DTABLE file.
• DTABLE_HEADER_READ reads the header from a DTABLE file.
• 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.
• MONOMIAL_QUADRATURE_GEN_LAGUERRE applies a quadrature rule to a monomial.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• 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.
• TIMESTRING writes the current YMDHMS date into a string.

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