HYPERCUBE_EXACTNESS
Exactness of Multidimensional Quadrature

HYPERCUBE_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule over the unit hypercube in M dimensions.

The polynomial exactness of a quadrature rule is defined as the highest total degree D such that the quadrature rule is guaranteed to integrate exactly all polynomials of total degree DEGREE_MAX or less, ignoring roundoff. The total degree of a polynomial is the maximum of the degrees of all its monomial terms. The degree of a monomial term is the sum of the exponents. Thus, for instance, the DEGREE of

x²y z⁵

To be thorough, the program starts at DEGREE = 0, and then proceeds to DEGREE = 1, 2, and so on up to a maximum degree DEGREE_MAX specified by the user. At each value of DEGREE, the program generates every possible monomial term, applies the quadrature rule to it, and determines the quadrature error. The program uses a scaling factor on each monomial so that the exact integral should always be 1; therefore, each reported error can be compared on a fixed scale.

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

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

• prefix_x.txt
• prefix_w.txt
• prefix_r.txt

The exactness results are written to an output file with the corresponding name:

• prefix_exact.txt

Usage:

hypercube_exactness ( 'prefix', degree_max )

• 'prefix' is the common prefix for the files containing the abscissa, weight and region 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 computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

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

hypercube_exactness_test

HYPERCUBE_GRID, a MATLAB library which computes a grid of points over the interior of a hypercube in M dimensions.

PYRAMID_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule over the interior of the unit pyramid in 3D.

SPHERE_EXACTNESS, a MATLAB program which tests the polynomial exactness of a quadrature rule over the surface of the unit sphere in 3D.

SQUARE_EXACTNESS, a MATLAB library which investigates the polynomial exactness of quadrature rules for f(x,y) over the interior of a square (rectangle/quadrilateral) in 2D.

TETRAHEDRON_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

TRIANGLE_EXACTNESS, a MATLAB program which investigates the monomial exactness quadrature rule over the interior of a triangle in 2D.

WEDGE_EXACTNESS, a MATLAB program which investigates the monomial exactness of a quadrature rule over the interior of the unit wedge in 3D.

Reference:

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

Source Code:

• hypercube_exactness.m, is the source code.
• timestamp.m, prints the YMDHMS date as a timestamp.