NCC_TRIANGLE is a FORTRAN90 library which defines the weights and abscisass for a sequence of 9 Newton-Cotes closed 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 "closed" when they 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.
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 8.
The computer code and data files made available on this web page are distributed under the GNU LGPL license.
NCC_TRIANGLE is available in a C++ version and a FORTRAN90 version and a MATLAB version
DUNAVANT, a FORTRAN90 library which defines Dunavant rules for quadrature on a triangle.
FEKETE, a FORTRAN90 library which defines Fekete rules for quadrature or interpolation on a triangle.
FELIPPA, a FORTRAN90 library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.
GM_RULE, a FORTRAN90 library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.
LYNESS_RULE, a FORTRAN90 library which returns Lyness-Jespersen quadrature rules for the triangle.
NCC_TETRAHEDRON, a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a tetrahedron.
NCO_TETRAHEDRON, a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a tetrahedron.
NCO_TRIANGLE, a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a triangle.
NINTLIB, a FORTRAN90 library which defines a variety of routines for numerical estimation of integrals in multiple dimensions.
QUADRATURE_RULES_TRI, a dataset directory which contains triples of files which defines various quadrature rules on triangles.
QUADRULE, a FORTRAN90 library which defines quadrature rules on a variety of intervals with different weight functions.
STROUD, a FORTRAN90 library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.
TEST_TRI_INT, a FORTRAN90 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.
TOMS706, a FORTRAN77 library which estimates the integral of a function over a triangulated region.
TRIANGLE_EXACTNESS, a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the triangle.
TRIANGLE_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over a triangle.
WANDZURA, a FORTRAN90 library which defines Wandzura rules for quadrature on a triangle.
One of the tests in the sample calling program creates EPS files of the abscissas in the unit triangle. These have been converted to PNG files for display here.
You can go up one level to the FORTRAN90 source codes.