chebyshev2_exactness

chebyshev2_exactness, a Python code which investigates the polynomial exactness of a Gauss-Chebyshev type 2 quadrature rule for the interval [-1,+1].

Standard Gauss-Chebyshev type 2 quadrature assumes that the integrand we are considering has a form like:

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

A standard Gauss-Chebyshev type 2 quadrature rule is a set of n positive weights w and abscissas x so that

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

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

For a standard Gauss-Chebyshev type 2 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 ) f(x) * sqrt ( 1 - x^2 ) 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.

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

The exactness results are written to an output file with the corresponding name:

• prefix_exact.txt

Usage:

chebyshev2_exactness ( 'prefix', degree_max )

• '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.

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 MIT license

Languages:

chebyshev2_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:

chebyshev_polynomial, a Python code which evaluates the Chebyshev polynomial and associated functions.

chebyshev2_rule, a Python code which generates a Gauss-Chebyshev type 2 quadrature rule.

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

Source Code:

• chebyshev2_exactness.py, the source code.
• chebyshev2_exactness.sh, runs all the tests.
• chebyshev2_exactness.txt, the output file.

CHEBY2_O4 is a standard Gauss-Chebyshev type 2 order 4 rule.

• cheby2_o4_x.txt, the abscissas of the rule.
• cheby2_o4_w.txt, the weights of the rule.
• cheby2_o4_r.txt, defines the region for the rule.