NINT_EXACTNESS_MIXED
Exactness of Multidimensional Quadrature Using Mixed Rules


NINT_EXACTNESS_MIXED is a MATLAB program which investigates the polynomial exactness of a multidimensional quadrature rule based on a direct product of one-dimensional rules. The one dimensional rules, in turn, may be any mixture of rules for 6 standard quadrature problems.

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:

INT_EXACTNESS, a MATLAB program which tests the polynomial exactness of one dimensional quadrature rules.

NINT_EXACTNESS, a MATLAB program which tests the polynomial exactness of integration rules for the unit hypercube.

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

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

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

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

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

TETRAHEDRON_EXACTNESS, a MATLAB 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:

Examples and Tests:

SPARSE_GRID_MIXED_D2_L2_CCXCC is a level 2 sparse grid quadrature rule for dimension 2, based on 1D factors of [Clenshaw Curtis, Clenshaw Curtis].

SPARSE_GRID_MIXED_D2_L2_CCXGL is a level 2 sparse grid quadrature rule for dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Legendre].

SPARSE_GRID_MIXED_D2_L2_CCXGLG is a level 2 sparse grid quadrature rule for dimension 2, based on 1D factors of [Clenshaw Curtis, Generalized Gauss Laguerre]. The Generalized Gauss Laguerre rule uses ALPHA = 1.5.

SPARSE_GRID_MIXED_D2_L2_CCXLG is a level 2 sparse grid quadrature rule for dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Laguerre].

SPARSE_GRID_MIXED_D2_L2_F2XGJ is a level 2 sparse grid quadrature rule for dimension 2, based on 1D factors of [Fejer Type 2, Gauss Jacobi]. The Gauss Jacobi rule uses ALPHA = 0.5, BETA = 1.5.

SPARSE_GRID_MIXED_D2_L3_CCXGP is a level 3 sparse grid quadrature rule for dimension 2, based on 1D factors of [Clenshaw Curtis, Gauss Patterson].

SPARSE_GRID_MIXED_D3_L2_CCXF2XGH is a level 2 sparse grid quadrature rule for dimension 3, based on 1D factors of [Clenshaw Curtis, Fejer Type 2, Gauss Hermite].

You can go up one level to the MATLAB source codes.


Last revised on 17 October 2008.