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.

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

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:

ANNULUS_RULE, a C code which computes a quadrature rule for estimating integrals of a function over the interior of a circular annulus in 2D.

CUBE_FELIPPA_RULE, a C code which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

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

PYRAMID_FELIPPA_RULE, a C code which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SIMPLEX_GM_RULE, a C code which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

SQUARE_FELIPPA_RULE, a C code which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

STROUD, a C code which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TETRAHEDRON_FELIPPA_RULE, a C code which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

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

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

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

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

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

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

triangle_nco_rule_test

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.

WEDGE_FELIPPA_RULE, a C code which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

Reference:

  1. Peter Silvester,
    Symmetric Quadrature Formulae for Simplexes,
    Mathematics of Computation,
    Volume 24, Number 109, January 1970, pages 95-100.

Source Code:


Last revised on 22 August 2019.