exactness

exactness, a MATLAB 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 information on this web page is 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 an Octave version and a Python version.

Related Data and Programs:

exactness_test

chebyshev1_exactness, a MATLAB code which tests the monomial exactness of Gauss-Chebyshev type 1 quadrature rules.

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

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

gen_laguerre_exactness, a MATLAB code which tests the exactness of generalized Gauss-Laguerre quadrature rules.

hermite_exactness, a MATLAB code which tests the 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 code 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 code 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].

Reference:

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

Source Code:

• chebyshev1_exactness.m, monomial exactness for the Chebyshev1 integral.
• chebyshev1_integral.m, evaluates a monomial Chebyshev1 integral.
• chebyshev1_set.m, sets a Chebyshev Type 1 rule.
• chebyshev2_exactness.m, monomial exactness for the Chebyshev2 integral.
• chebyshev2_integral.m, evaluates a monomial Chebyshev2 integral.
• chebyshev2_set.m, sets a Chebyshev Type 2 rule.
• chebyshev3_set.m, sets a Chebyshev Type 3 rule.
• clenshaw_curtis_set.m, sets a Clenshaw-Curtis rule.
• fejer1_set.m, sets a Fejer type 1 rule.
• fejer2_set.m, sets a Fejer type 2 rule.
• gegenbauer_exactness.m, monomial exactness for the Gegenbauer integral.
• gegenbauer_integral.m, evaluates a monomial Gegenbauer integral.
• hermite_exactness.m, monomial exactness for the Hermite integral.
• hermite_integral.m, evaluates a monomial Hermite integral.
• hermite_set.m, sets a Gauss-Hermite rule for density exp(-x^2).
• hermite_1_set.m, sets a Gauss-Hermite rule for density 1.
• laguerre_exactness.m, monomial exactness for the Laguerre integral.
• laguerre_integral.m, evaluates a monomial Laguerre integral.
• laguerre_set.m, sets a Gauss-Laguerre rule for density exp(-x).
• laguerre_1_set.m, sets a Gauss-Laguerre rule for density 1.
• legendre_exactness.m, monomial exactness for the Legendre integral.
• legendre_integral.m, evaluates a monomial Legendre integral.
• legendre_set.m, sets a Gauss-Legendre rule for density 1.
• r8_factorial2.m, evaluates the double factorial function.