TETRAHEDRON_EXACTNESS is a MATLAB program which investigates the polynomial exactness of a quadrature rule for a tetrahedron.
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
x2yz5is 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
For information on the form of these files, see the QUADRATURE_RULES_TET directory listed below.
The exactness results are written to an output file with the corresponding name:
tetrahedron_exactness ( 'prefix', degree_max )where
If the arguments are not supplied on the command line, the program will prompt for them.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
TETRAHEDRON_EXACTNESS is available in a C++ version and a FORTRAN90 version and a MATLAB version.
GM_RULE, a MATLAB library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.
INTEGRAL_TEST, a FORTRAN90 program which uses test integrals to measure the effectiveness of certain sets of quadrature rules.
KEAST, a MATLAB library which sets up a Keast quadrature rule for a tetrahedron.
NCC_TETRAHEDRON, a MATLAB library which sets up a Newton Cotes Closed quadrature rule for a tetrahedron.
NCO_TETRAHEDRON, a MATLAB library which sets up a Newton Cotes Open quadrature rule for a tetrahedron.
NINT_EXACTNESS, a MATLAB program which tests the exactness of integration rules over the interval, rectangle, or multidimensional rectangle.
NINT_EXACTNESS_MIXED, a MATLAB program which measures the polynomial exactness of a multidimensional quadrature rule based on a mixture of 1D quadrature rule factors.
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.
QUADRATURE_RULES_TET, a dataset directory which contains sets of files that define quadrature rules over a tetrahedron.
QUADRULE, a MATLAB library which defines quadrature rules on a variety of intervals with different weight functions.
SPHERE_EXACTNESS, a MATLAB program which tests the polynomial exactness of a quadrature rule for the unit sphere;
STROUD, a MATLAB library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and multiple dimensions.
TETRAHEDRON_MONTE_CARLO, a MATLAB program which uses the Monte Carlo method to estimate integrals over a tetrahedron.
TRIANGLE_EXACTNESS, a MATLAB program which investigates the polynomial exactness of a quadrature rule for the triangle.
The "W", "X" and "R" files associated with the point sets tested here may be found in the QUADRATURE_RULES_TET directory.
You can go up one level to the MATLAB source codes.