NCC_TETRAHEDERON is a MATLAB library which defines the weights and abscisass for a sequence of 7 Newton-Cotes closed quadrature rules for the tetrahedron.
Newton-Cotes rules have the characteristic that the abscissas are equally spaced. For a tetrahedron, this refers to spacing in the unit reference tetrahedron, or in the barycentric coordinate system. These rules may be mapped to an arbitrary tetrahedron, and will still be valid.
The rules are said to be "closed" when they include points on the boundary of the tetrahedron.
The use of equally spaced abscissas may be important for your application. That may how your data was collected, for instance. On the other hand, the use of equally spaced abscissas carries a few costs. In particular, for a given degree of polynomial accuracy, there will be rules that achieve this accuracy, but use fewer abscissas than Newton-Cotes. Moreover, the Newton-Cotes approach almost always results in negative weights for some abscissas. This is generally an undesirable feature, particularly when higher order quadrature rules are being used.
(Note that the first rule included in the set is not, strictly speaking, a Newton-Cotes closed rule; it's just the rule that uses a single point at the centroid. However, by including this rule as the first in the set, we have a rule with each polynomial degree of exactness from 0 to 6.)
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
NCC_TETRAHEDRON is available in a C++ version and a FORTRAN90 version and a MATLAB version
FELIPPA, a MATLAB library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.
GM_RULE, a MATLAB library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.
KEAST, a MATLAB library which defines a number of quadrature rules for a tetrahedron.
NCC_TRIANGLE, a MATLAB library which defines Newton-Cotes closed quadrature rules on a triangle.
NCO_TETRAHEDRON, a MATLAB library which defines Newton-Cotes open quadrature rules on a tetrahedron.
NINTLIB, a MATLAB library which contains a variety of routines for numerical estimation of integrals in multiple dimensions.
QUADRATURE_RULES_TET, a dataset directory which contains triples of files which define various quadrature rules on tetrahedrons.
QUADRULE, a MATLAB library which defines quadrature rules on a variety of intervals with different weight functions.
STROUD, a MATLAB library which contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.
TET_MESH, a MATLAB library which is useful for work with tetrahedral meshes in 3D;
TETRAHEDRON_MONTE_CARLO, a MATLAB program which uses the Monte Carlo method to estimate integrals over a tetrahedron.
TETRAHEDRONS, a dataset directory which contains examplesof tetrahedrons;
You can go up one level to the MATLAB source codes.