NINT_EXACTNESS_MIXED
Exactness of Multidimensional Quadrature Using Mixed Rules


NINT_EXACTNESS_MIXED, a C++ program which investigates the polynomial exactness of a multidimensional quadrature rule which is designed for a quadrature region that is a direct product of 1D regions which are a mixture of Legendre, Laguerre, and Hermite type regions.

The six standard 1d quadrature problems that may be used as factors for the multidimensional problem are:

The M-dimensional quadrature region R based on mixed factors is formed by the direct product

R = R1 x R2 x ... x Rm
where each factor region Ri is the region associated with one of the six rules. Thus, R is a sort of generalized hyperrectangle, with the understanding that in some coordinate directions the region may be semi-infinite or infinite.

The M-dimensional weight function W based on mixed factors is formed by the dproduct

w(x1,x2,...xm) = w1(x1) x w2(x2) x ... x wm(xm)
where each factor weight wi(xi) is the weight function associated with one of the six rules. Some weight functions include parameters alpha and beta, and these parameters may be specified independently in each dimension.

For a quadrature region R based on mixed factors, the corresponding monomial integrand has the form

Mono(X,E) = X1^E1 x X2^E2 x ... x Xm^Em
where each exponent Ei is a nonnegative integer.

The total degree of a monomial Mono(X,E) is:

TotalDegree(Mono(X,E)) = Sum ( 1 <= I <= M ) E(I)

Thus, for instance, the total degree of

x12 * x2 * x35
is 2+1+5=8.

The corresponding monomial integral is:

Integral ( X in R ) Mono(X,E) W(X) dX
where each exponent Ei is a nonnegative integer.

The monomial exactness of a quadrature rule is the maximum number D such that, for every monomial of total degree D or less, the quadrature rule produces the exact value of the monomial integral.

The polynomial exactness of a quadrature rule is the maximum number D such that, for every polynomial of total degree D or less, the quadrature rule produces the exact value of the polynomial integral. The total degree of a polynomial is simply the maximum of the total degrees of the monomials that form the polynomial.

This program is given a quadrature rule based on mixed factors, and seeks to determine the polynomial exactness of the rule. It does this simply by applying the quadrature rule to all the monomials of a total degree 0 up to some limit specified by the user.

The program is very flexible and interactive. The quadrature rule is defined by five files, to be read at input, and the maximum degree is specified by the user as well.

The files that define the quadrature rule are assumed to have related names, of the form

When running the program, the user only enters the common prefix part of the file names, which is enough information for the program to find all the files.

Usage:

nint_exactness_mixed prefix degree_max
where

If the arguments are not supplied on the command line, the program will prompt for them.

Licensing:

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

Languages:

NINT_EXACTNESS_MIXED is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

nint_exactness_mixed_test

NINTLIB, a C++ library which numerically estimates integrals in multiple dimensions.

PYRAMID_EXACTNESS, a C++ program which investigates the polynomial exactness of a quadrature rule for the pyramid.

SPARSE_GRID_MIXED, a dataset directory which contains multidimensional Smolyak sparse grids based on a mixed set of 1D factor rules.

SPARSE_GRID_MIXED, a C++ library which defines multidimensional quadrature rules using sparse grid techniques, based on a mixed set of 1D quadrature rules.

SPHERE_EXACTNESS, a C++ program which tests the polynomial exactness of a quadrature rule for the unit sphere;

STROUD, a C++ library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and multiple dimensions.

TEST_NINT, a C++ library which defines integrand functions for testing multidimensional quadrature routines.

TESTPACK, a C++ library which defines a set of integrands used to test multidimensional quadrature.

TETRAHEDRON_EXACTNESS, a C++ program which investigates the polynomial exactness of a quadrature rule for the tetrahedron.

Reference:

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

Source Code:

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Last revised on 28 March 2020.