legendre_exactness

legendre_exactness, a Python code which investigates the polynomial exactness of a Gauss-Legendre quadrature rule for the interval [-1,+1].

This program is actually appropriate for any quadrature rule that estimates integrals on [-1,+1], and which does not including a weighting function w(x) in the integral. This includes:

• Clenshaw-Curtis rules;
• Fejer rules of Type 1 or 2;
• Gauss-Legendre rules;
• Gauss-Lobatto rules (Gauss rule including both endpoints);
• Gauss-Patterson rules;
• Gauss-Radau rules (Gauss rule including one endpoint);
• Newton-Cotes rules, open and closed forms;

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

        Integral ( -1 <= x <= +1 ) f(x) dx
      

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

        Integral ( -1 <= x <= +1 ) f(x) dx
      

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

For a standard Gauss-Legendre 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) 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:

legendre_exactness ( 'prefix', degree_max )

• 'prefix' is the common prefix for the files containing the abscissa, weight and region information of the quadrature rule, entered as a quoted string;
• degree_max is the maximum monomial degree to check. This would normally be a relatively small nonnegative number, such as 5, 10 or 15.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

legendre_exactness is available in a C version and 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 monomial exactness of Gauss-Chebyshev type 1 quadrature rules.

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

exactness, a Python code which investigates the exactness of quadrature rules that estimate the integral of a function with a density, such as 1, exp(-x) or exp(-x^2), over an interval such as [-1,+1], [0,+oo) or (-oo,+oo).

gegenbauer_exactness, a Python code which tests the monomial exactness of Gauss-Gegenbauer quadrature rules.

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_rule, a Python code which generates a Gauss-Legendre quadrature rule.

Reference:

1. Philip Davis, Philip Rabinowitz,
   Methods of Numerical Integration,
   Second Edition,
   Dover, 2007,
   ISBN: 0486453391,
   LC: QA299.3.D28.

Source Code:

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

LEG_O004 is a standard Gauss-Legendre order 4 rule.

• leg_o004_x.txt, the abscissas of the rule.
• leg_o004_w.txt, the weights of the rule.
• leg_o004_r.txt, defines the region for the rule.