GEGENBAUER_EXACTNESS
Exactness of Gauss-Gegenbauer Quadrature Rules

GEGENBAUER_EXACTNESS is a C++ 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:

gegenbauer_exactness prefix degree_max alpha

• 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.
• 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

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

Related Data and Programs:

GEGENBAUER_CC, a C++ library which estimates the Gegenbauer weighted integral of a function f(x) using a Clenshaw-Curtis approach.

GEGENBAUER_RULE, a C++ program which can generate a Gauss-Gegenbauer quadrature rule on request.

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

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

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

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

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

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

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

LAGUERRE_EXACTNESS, a C++ program which tests the polynomial exactness of Gauss-Laguerre quadrature rules for integration over [0,+oo) with density function exp(-x).

LEGENDRE_EXACTNESS, a C++ program which tests the monomial exactness of quadrature rules for the Legendre problem of integrating a function with density 1 over the interval [-1,+1].

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:

• gegenbauer_exactness.cpp, 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

  gegenbauer_exactness 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

  gegenbauer_exactness 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

  gegenbauer_exactness 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

  gegenbauer_exactness 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

  gegenbauer_exactness gegen_o16_a0.5 35 0.5

List of Routines:

• MAIN is the main program for GEGENBAUER_EXACTNESS.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• DTABLE_DATA_READ reads the 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.
• GEGENBAUER_INTEGRAL evaluates the integral of a monomial with Gegenbauer weight.
• MONOMIAL_QUADRATURE_GEGENBAUER applies a quadrature rule to a monomial.
• R8_ABS returns the absolute value of an R8.
• R8_GAMMA evaluates Gamma(X) for a real argument.
• R8_HUGE returns a "huge" R8.
• R8_HYPER_2F1 evaluates the hypergeometric function 2F1(A,B,C,X).
• R8_PSI evaluates the function Psi(X).
• S_LEN_TRIM returns the length of a string to the last nonblank.
• 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 returns the current YMDHMS date as a string.

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