TETRAHEDRON_NCO_RULE is a C library which defines the weights and abscisass for a sequence of 7 Newton-Cotes open quadrature rules over the interior of a tetrahedron in 3D.
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 "open" when they do not 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.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
TETRAHEDRON_NCO_RULE is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version
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TRIANGLE_NCO_RULE, a C library which defines Newton-Cotes open quadrature rules over the interior of a triangle in 2D.
WEDGE_FELIPPA_RULE, a C library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.
You can go up one level to the C++ source codes.