exactness

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
      

        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:

• Chebyshev Type 1 density 1/sqrt(1-x^2), over [-1,+1],
• Chebyshev Type 2 density sqrt(1-x^2), over [-1,+1].
• Gegenbauer density (1-x^2)^(lambda-1/2), over [-1,+1].
• Hermite (physicist) density 1/sqrt(pi) exp(-x^2), over (-oo,+oo).
• Hermite (probabilist) density 1/sqrt(2*pi) exp(-x^2/2), over (-oo,+oo).
• Hermite (unit) density 1, over (-oo,+oo).
• Jacobi density (1-x)^alpha*(1+x)^beta, over [-1,+1].
• Laguerre (standard) density exp(-1), over [0,+oo).
• Laguerre (unit) density 1, over [0,+oo).
• Legendre density 1, over [-1,+1].

Common quadrature rules include:

• Clenshaw-Curtis quadrature for Legendre density, exactness = n - 1;
• Fejer Type 1 quadrature for Legendre density, exactness = n - 1;
• Fejer Type 2 quadrature for Legendre density, exactness = n - 1;
• Gauss-Chebyshev Type 1 quadrature, exactness = 2 * n - 1;
• Gauss-Chebyshev Type 2 quadrature, exactness = 2 * n - 1;
• Gauss-Gegenbauer quadrature, exactness = 2 * n - 1;
• Gauss-Hermite quadrature, exactness = 2 * n - 1;
• Gauss-Laguerre quadrature, exactness = 2 * n - 1;
• Gauss-Legendre quadrature, exactness = 2 * n - 1;

Licensing:

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

Languages:

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.

Reference:

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

Source Code:

• exactness.py, calls all the test.
• exactness.sh, runs all the test.
• exactness.txt, the output file.