# NCO_TRIANGLE Newton-Cotes Open Quadrature Rules for the Triangle

NCO_TRIANGLE is a MATLAB library which defines the weights and abscisass for a sequence of 9 Newton-Cotes open quadrature rules for the triangle.

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:

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

### Related Data and Programs:

DUNAVANT, a MATLAB library which defines Dunavant rules for quadrature on a triangle.

FEKETE, a MATLAB library which defines Fekete rules for quadrature or interpolation on a triangle.

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.

LYNESS_RULE, a MATLAB library which returns Lyness-Jespersen quadrature rules for the triangle.

NCC_TETRAHEDRON, a MATLAB library which defines Newton-Cotes closed quadrature rules on 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.

STROUD, a MATLAB library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and M-dimensions.

TEST_TRI_INT, a MATLAB library which can be used to test algorithms for quadrature over a triangle.

TOMS612, a FORTRAN77 library which can estimate the integral of a function over a triangle.

TRIANGLE_MONTE_CARLO, a MATLAB program which uses the Monte Carlo method to estimate integrals over a triangle.

WANDZURA, a MATLAB library which defines Dunavant rules for quadrature on a triangle.

### Reference:

1. Peter Silvester,