# gegenbauer_exactness

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

• 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:

gegenbauer_exactness ( 'prefix', degree_max, alpha )
where
• '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.

### Languages

gegenbauer_exactness is available in a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.

### Related Data and Programs:

chebyshev1_exactness, a Python code which tests the polynomial exactness of Gauss-Chebyshev type 1 quadrature rules.

chebyshev2_exactness, a Python code which tests the polynomial exactness of Gauss-Chebyshev type 2 quadrature rules.

exactness, a Python code which tests the polynomial exactness of a quadrature rule for a finite interval.

gegenbauer_cc, a Python code which estimates the Gegenbauer weighted integral of a function f(x) using a Clenshaw-Curtis approach.

gegenbauer_polynomial, a Python code which evaluates the Gegenbauer polynomial and associated functions.

gegenbauer_rule, a Python code which can generate a Gauss-Gegenbauer quadrature rule on request.

gen_hermite_exactness, a Python code which tests the polynomial exactness of generalized Gauss-Hermite quadrature rules.

gen_laguerre_exactness, a Python code which tests the polynomial exactness of generalized Gauss-Laguerre quadrature rules.

hermite_exactness, a Python code which tests the polynomial exactness of Gauss-Hermite quadrature rules.

jacobi_exactness, a Python code which tests the polynomial exactness of Gauss-Jacobi quadrature rules.

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

legendre_exactness, a Python code 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:

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

Last revised on 19 May 2023.