gegenbauer_exactness, a Python code 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
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:
gegenbauer_exactness ( 'prefix', degree_max, alpha )where
If the arguments are not supplied on the command line, the program will prompt for them.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
gegenbauer_exactness is available in a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.
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