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:

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 GNU LGPL 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:

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

exactness_test

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:

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_factorial.m, evaluates the factorial function.
r8_factorial2.m, evaluates the double factorial function.
timestamp.m, prints the YMDHMS date as a timestamp