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).

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:


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


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

Related Data and Programs:

quadrule, a Python code which defines quadrature rules for approximating an integral over a 1D domain.


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

Source Code:

Last revised on 23 January 2020.