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:
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) dxmay be approximated by
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
The exactness results are written to an output file with the corresponding name:
legendre_exactness ( 'prefix', degree_max )where
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
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.
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.
LEG_O004 is a standard Gauss-Legendre order 4 rule.