tetrahedron_nco_rule, a FORTRAN90 code 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.
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 MIT 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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