EXACTNESS
Exactness of Quadrature Rules


EXACTNESS is a MATLAB library 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).

A 1D quadrature rule estimates I(f), the integral of a function f(x) over an interval [a,b] with density rho(x):

        I(f) = integral ( a < x < b ) f(x) rho(x) dx
      
by a n-point quadrature rule of weights w and points x:
        Q(f) = sum ( 1 <= i <= n ) w(i) f(x(i))
      

Most quadrature rules come in a family of various sizes. A quadrature rule of size n is said to have exactness p if it is true that the quadrature estimate is exactly equal to the exact integral for every monomial (and hence, polynomial) whose degree is p or less.

This program allows the user to specify a quadrature rule, a size n, and a degree p_max. It then computes and compares the exact integral and quadrature estimate for monomials of degree 0 through p_max, so that the user can analyze the results.

Common densities include:

Common quadrature rules include:

Licensing:

The computer code and data files made available on this web page are distributed under the GNU LGPL license.

Languages:

EXACTNESS is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

CUBE_EXACTNESS, a MATLAB library which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3D.

HERMITE_EXACTNESS, a MATLAB program which tests the monomial exactness of Gauss-Hermite quadrature rules for estimating the integral of a function with density exp(-x^2) over the interval (-oo,+oo).

LAGUERRE_EXACTNESS, a MATLAB program which tests the monomial exactness of Gauss-Laguerre quadrature rules for estimating the integral of a function with density exp(-x) over the interval [0,+oo).

LEGENDRE_EXACTNESS, a MATLAB program which tests the monomial exactness of Gauss-Legendre quadrature rules for estimating the integral of a function with density 1 over the interval [-1,+1].

SQUARE_EXACTNESS, a MATLAB library which investigates the exactness of 2D quadrature rules that estimate the integral of a function f(x,y) over a 2D domain.

Reference:

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

Source Code:

Examples and Tests:

You can go up one level to the MATLAB source codes.


Last revised on 11 January 2016.