# tetrahedron_exactness

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

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

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

• prefix_exact.txt

### Usage:

tetrahedron_exactness ( 'prefix', degree_max )
where
• '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.

### Languages:

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.

### Related Data and Programs:

cube_exactness, a Python code which investigates the polynomial exactness of quadrature rules over the interior of a cube in 3d.

hypercube_exactness, a Python code which measures the monomial exactness of an m-dimensional quadrature rule over the interior of the unit hypercube in m dimensions.

pyramid_exactness, a Python code which investigates the polynomial exactness of a quadrature rule over the interior of the pyramid in 3d.

sphere_exactness, a Python code which tests the polynomial exactness of a quadrature rule over the surface of the unit sphere in 3d;

square_exactness, a Python code which investigates the polynomial exactness of quadrature rules for f(x,y) over the interior of a square (rectangle/quadrilateral) in 2d.

tetrahedron, a Python code which carries out geometric calculations involving a general tetrahedron, including solid and facial angles, face areas, point containment, distances to a point, circumsphere and insphere, measures of shape quality, centroid, barycentric coordinates, edges and edge lengths, random sampling, and volumes.

tetrahedron_arbq_rule, a Python code 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 Python code which returns a Felippa quadrature rule for approximating integrals over the interior of a tetrahedron in 3d.

tetrahedron_integrals, a Python code which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3d.

tetrahedron_jaskowiec_rule, a Python code which returns quadrature rules, with exactness up to total degree 20, over the interior of a tetrahedron in 3D, by Jan Jaskowiec, Natarajan Sukumar.

tetrahedron_keast_rule, a Python code which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3d.

tetrahedron_monte_carlo, a Python code which uses the Monte Carlo method to estimate integrals over the interior of a tetrahedron in 3d.

tetrahedron_ncc_rule, a Python code which defines Newton-Cotes closed (NCC) quadrature rules over the interior of a tetrahedron in 3d.

tetrahedron_nco_rule, a Python code which defines Newton-Cotes open (NCO) quadrature rules over the interior of a tetrahedron in 3d.

tetrahedron_witherden_rule, a Python code which returns a symmetric Witherden quadrature rule for the tetrahedron, with exactness up to total degree 10.

triangle_exactness, a Python code which investigates the monomial exactness quadrature rule over the interior of a triangle in 2d.

wedge_exactness, a Python code 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:

Last revised on 23 May 2023.