# triangle_nco_rule

triangle_nco_rule, a C code which defines the weights and abscisass for a sequence of 9 Newton-Cotes open quadrature rules over the interior of a triangle in 2D.

Newton-Cotes rules have the characteristic that the abscissas are equally spaced. For a triangle, this refers to spacing in the unit reference triangle, or in the barycentric coordinate system. These rules may be mapped to an arbitrary triangle, and will still be valid.

The rules are said to be "open" when they do not include points on the boundary of the triangle.

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.

### Languages:

triangle_nco_rule is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version

### Related Data and Programs:

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TRIANGLE_MONTE_CARLO, a C code which uses the Monte Carlo method to estimate integrals over the interior of a triangle in 2D.

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TRIANGLE_SYMQ_RULE, a C code which returns efficient symmetric quadrature rules, with exactness up to total degree 50, over the interior of an arbitrary triangle in 2D, by Hong Xiao and Zydrunas Gimbutas.

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### Reference:

1. Peter Silvester,