TETRAHEDRON_NCO_RULE
Newton-Cotes Open Quadrature for the Tetrahedron


TETRAHEDRON_NCO_RULE, a FORTRAN90 code which defines the weights and abscisass for a sequence of 7 Newton-Cotes open quadrature rules over the interior of a tetrahedron in 3D.

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

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

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 6.

Licensing:

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

Languages:

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

Related Data and Programs:

CUBE_FELIPPA_RULE, a FORTRAN90 code which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

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.

PYRAMID_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a pyramid in 3D.

SIMPLEX_GM_RULE, a FORTRAN90 code which defines Grundmann-Moeller quadrature rules over the interior of a simplex in M dimensions.

SQUARE_FELIPPA_RULE, a FORTRAN90 code which returns the points and weights of a Felippa quadrature rule over the interior of a square in 2D.

TETRAHEDRON_ARBQ_RULE, a FORTRAN90 code which returns quadrature rules, with exactness up to total degree 15, over the interior of a tetrahedron in 3D, by Hong Xiao and Zydrunas Gimbutas.

TETRAHEDRON_EXACTNESS, a FORTRAN90 code which investigates the polynomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

TETRAHEDRON_FELIPPA_RULE, a FORTRAN90 code which returns Felippa's quadratures rules for approximating integrals over the interior of a tetrahedron in 3D.

TETRAHEDRON_INTEGRALS, a FORTRAN90 code which returns the exact value of the integral of any monomial over the interior of the unit tetrahedron in 3D.

TETRAHEDRON_KEAST_RULE, a FORTRAN90 code which defines ten quadrature rules, with exactness degrees 0 through 8, over the interior of a tetrahedron in 3D.

TETRAHEDRON_MONTE_CARLO, a MATLAB program which uses the Monte Carlo method to estimate integrals over the interior of a tetrahedron in 3D.

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

tetrahedron_nco_rule_test

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

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 08 September 2020.