tetrahedron_exactness, a Python code 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
x^{2}yz^{5}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
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 MIT license
tetrahedron_exactness is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version and a Python version.
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