TETRAHEDRON_EXACTNESS
Exactness of Quadrature on Tetrahedrons


TETRAHEDRON_EXACTNESS is a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

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. For a tetrahedron, the degree of a monomial term is the sum of the exponents of x, y and z. Thus, for instance, the DEGREE of

x2yz5
is 2+1+5=8.

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

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 three files.

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

Usage:

tetrahedron_exactness 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:

TETRAHEDRON_EXACTNESS is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

CUBE_EXACTNESS, a FORTRAN90 library which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3D.

HYPERCUBE_EXACTNESS, a FORTRAN90 program which measures the monomial exactness of an M-dimensional quadrature rule over the interior of the unit hypercube in M dimensions.

PYRAMID_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule over the interior of the unit pyramid in 3D.

SPHERE_EXACTNESS, a FORTRAN90 program which tests the polynomial exactness of a quadrature rule over the surface of the unit sphere in 3D.

SQUARE_EXACTNESS, a FORTRAN90 library which investigates the polynomial exactness of quadrature rules for f(x,y) over the interior of a rectangle in 2D.

TETRAHEDRON_ARBQ_RULE, a FORTRAN90 library which returns quadrature rules, with exactness up to total degree 15, over the interior of a tetrahedron in 3D, by Hong Xiao and Zydrunas Gimbutas.

TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 library which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TETRAHEDRON_INTEGRALS, a FORTRAN90 library which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TETRAHEDRON_KEAST_RULE, a FORTRAN90 library which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

TETRAHEDRON_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over a tetrahedron.

TETRAHEDRON_NCC_RULE, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

TETRAHEDRON_NCO_RULE, a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a tetrahedron.

TRIANGLE_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the triangle.

WEDGE_EXACTNESS, a FORTRAN90 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:

Examples and Tests:

The "W", "X" and "R" files associated with the point sets tested here may be found in the QUADRATURE_RULES_TET directory.

List of Routines:

You can go up one level to the FORTRAN90 source codes.


Last revised on 14 October 2009.