# triangle_nco_rule

triangle_nco_rule, a FORTRAN90 code which defines the weights and abscisass for a sequence of 9 Newton-Cotes open quadrature rules over the interior of the 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:

LINE_NCO_RULE, a FORTRAN90 code which computes a Newton Cotes Open (NCO) quadrature rule, using equally spaced points, over the interior of a line segment in 1D.

TETRAHEDRON_NCO_RULE, a FORTRAN90 code which defines Newton-Cotes open quadrature rules over the interior of a tetrahedron in 3D.

TRIANGLE_DUNAVANT_RULE, a FORTRAN90 code which sets up a Dunavant quadrature rule over the interior of a triangle in 2D.

TRIANGLE_EXACTNESS, a FORTRAN90 code which investigates the polynomial exactness of a quadrature rule over the interior of a triangle in 2D.

TRIANGLE_FEKETE_RULE, a FORTRAN90 code which defines Fekete rules for interpolation or quadrature over the interior of a triangle in 2D.

TRIANGLE_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a triangle in 2D.

TRIANGLE_LYNESS_RULE, a FORTRAN90 code which returns Lyness-Jespersen quadrature rules over the interior of a triangle in 2D.

TRIANGLE_MONTE_CARLO, a FORTRAN90 code which uses the Monte Carlo method to estimate integrals over a triangle.

TRIANGLE_NCC_RULE, a FORTRAN90 code which defines Newton-Cotes closed quadrature rules over the interior of a triangle in 2D.

TRIANGLE_SYMQ_RULE, a FORTRAN90 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.

TRIANGLE_WANDZURA_RULE, a FORTRAN90 code which sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30 over the interior of a triangle in 2D.

### Reference:

1. Peter Silvester,