INT_EXACTNESS_GEGENBAUER
Exactness of Gauss-Gegenbauer Quadrature Rules

INT_EXACTNESS_GEGENBAUER is a MATLAB program which investigates the polynomial exactness of a Gauss-Gegenbauer quadrature rule for the interval [-1,1] with a weight function.

The Gauss-Gegenbauer quadrature rule is designed to approximate integrals on the interval [-1,1], with a weight function of the form (1-x^2)^ALPHA. ALPHA is a real parameter that must be greater than -1.

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

        Integral ( -1 <= x <= +1 ) (1-x^2)^alpha f(x) dx
      

For a Gauss-Gegenbauer 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 ( -1 <= x <= +1 ) (1-x^2)^alpha 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 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_gegenbauer ( 'prefix', degree_max, alpha )

• '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 exponent of (1-x^2) in the weight function; alpha should be a real number greater than -1.0.

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_GEGENBAUER is available in a C++ version and a FORTRAN90 version and a MATLAB version

Related Data and Programs:

GEGENBAUER_RULE, a MATLAB program which can generate a Gauss-Gegenbauer 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_GEN_HERMITE, a MATLAB program which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.

INT_EXACTNESS_GEN_LAGUERRE, a MATLAB program which tests the polynomial exactness of generalized Gauss-Laguerre 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.

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

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

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.
2. Shanjie Zhang, Jianming Jin,
   Computation of Special Functions,
   Wiley, 1996,
   ISBN: 0-471-11963-6,
   LC: QA351.C45.

Source Code:

• int_exactness_gegenbauer.m, is the source code.

Examples and Tests:

GEGEN_O1_A0.5 is a Gauss-Gegenbauer order 1 rule with ALPHA = 0.5.

• gegen_o1_a0.5_x.txt, the abscissas of the rule.
• gegen_o1_a0.5_w.txt, the weights of the rule.
• gegen_o1_a0.5_r.txt, defines the region for the rule.
• gegen_o1_a0.5_exact.txt, the results of the command

  int_exactness_gegenbauer ( 'gegen_o1_a0.5', 5, 0.5 )

GEGEN_O2_A0.5 is a Gauss-Gegenbauer order 2 rule with ALPHA = 0.5.

• gegen_o2_a0.5_x.txt, the abscissas of the rule.
• gegen_o2_a0.5_w.txt, the weights of the rule.
• gegen_o2_a0.5_r.txt, defines the region for the rule.
• gegen_o2_a0.5_exact.txt, the results of the command

  int_exactness_gegenbauer ( 'gegen_o2_a0.5', 5, 0.5 )

GEGEN_O4_A0.5 is a Gauss-Gegenbauer order 4 rule with ALPHA = 0.5.

• gegen_o4_a0.5_x.txt, the abscissas of the rule.
• gegen_o4_a0.5_w.txt, the weights of the rule.
• gegen_o4_a0.5_r.txt, defines the region for the rule.
• gegen_o4_a0.5_exact.txt, the results of the command

  int_exactness_gegenbauer ( 'gegen_o4_a0.5', 10, 0.5 )

GEGEN_O8_A0.5 is a Gauss-Gegenbauer order 8 rule with ALPHA = 0.5.

• gegen_o8_a0.5_x.txt, the abscissas of the rule.
• gegen_o8_a0.5_w.txt, the weights of the rule.
• gegen_o8_a0.5_r.txt, defines the region for the rule.
• gegen_o8_a0.5_exact.txt, the results of the command

  int_exactness_gegenbauer ( 'gegen_o8_a0.5', 18, 0.5 )

GEGEN_O16_A0.5 is a Gauss-Gegenbauer order 16 rule with ALPHA = 0.5.

• gegen_o16_a0.5_x.txt, the abscissas of the rule.
• gegen_o16_a0.5_w.txt, the weights of the rule.
• gegen_o16_a0.5_r.txt, defines the region for the rule.
• gegen_o16_a0.5_exact.txt, the results of the command

  int_exactness_gegenbauer ( 'gegen_o16_a0.5', 35, 0.5 )

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