nint_exactness_mixed

nint_exactness_mixed, a Fortran90 code 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:

Legendre integration
interval: [-1,+1],
weight: w(x) = 1,
rules: Clenshaw Curtis, Fejer Type 2, Gauss Legendre, Gauss Patterson.
Jacobi integration:
interval: [-1,+1],
weight: w(x) = (1-x)^alpha (1+x)^beta.
rules: Gauss Jacobi.
Laguerre integration
interval: [0,+oo),
weight: w(x) = exp(-x).
rules: Gauss Laguerre.
Generalized Laguerre integration
interval: [0,+oo),
weight: w(x) = x^alpha exp(-x).
rules: Gauss Laguerre.
Hermite integration
interval: (-oo,+oo),
weight: w(x) = exp(-x*x).
rules: Gauss Hermite.
Generalized Hermite integration
interval: (-oo,+oo),
weight: w(x) = |x|^alpha exp(-x*x).
rules: generalized Gauss Hermite.

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

prefix_a.txt, the "ALPHA" file;
prefix_b.txt, the "BETA" file;
prefix_r.txt, the "REGION" file;
prefix_w.txt, the "WEIGHT" file;
prefix_x.txt, the "ABSCISSA" file.

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

prefix is the common prefix for the files containing the alpha, beta, region, weight and abscissa information of the quadrature rule;
degree_max is the maximum total monomial degree to check. This should be a relatively small nonnegative number, particularly if the spatial dimension is high. A value of 5 or 10 might be reasonable, but a value of 50 or 100 is probably never a good input!

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

Licensing:

The information on this web page is distributed under the MIT license.

Languages:

nint_exactness_mixed is available in a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

Related Data and Programs:

nint_exactness_mixed_test

nintlib, a Fortran90 code which numerically estimates integrals in multiple dimensions.

pyramid_exactness, a Fortran90 code 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 Fortran90 code which defines multidimensional quadrature rules using sparse grid techniques, based on a mixed set of 1D quadrature rules.

sphere_exactness, a Fortran90 code which tests the polynomial exactness of a quadrature rule for the unit sphere;

stroud_rule, a Fortran90 code which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and multiple dimensions.

test_nint, a Fortran90 code which defines integrand functions for testing multidimensional quadrature routines.

testpack, a Fortran90 code which defines a set of integrands used to test multidimensional quadrature.

tetrahedron_exactness, a Fortran90 code which investigates the polynomial exactness of a quadrature rule for the tetrahedron.

Reference:

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

Source Code:

nint_exactness_mixed.f90, the source code.
nint_exactness_mixed.sh, compiles the source code.

Last revised on 28 September 2024.