TETRAHEDRON_NCC_RULE Newton-Cotes Closed Quadrature for the Tetrahedron

TETRAHEDRON_NCC_RULE is a C++ library which defines the weights and abscisass for a sequence of 7 Newton-Cotes closed 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 "closed" when they 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_NCC_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++ library 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++ library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

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

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

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

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

TETRAHEDRON_ARBQ_RULE, a C++ library 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 C++ program which investigates the monomial exactness of a quadrature rule over the interior of a tetrahedron in 3D.

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

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

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

TETRAHEDRON_MONTE_CARLO, a C++ library which uses the Monte Carlo method to estimate integrals over the interior of the unit tetrahedron in 3D.

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

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

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

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

WEDGE_FELIPPA_RULE, a C++ library 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.

List of Routines:

• R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT.
• REFERENCE_TO_PHYSICAL_T4 maps T4 reference points to physical points.
• TETRAHEDRON_NCC_DEGREE: degree of an NCC rule for the tetrahedron.
• TETRAHEDRON_NCC_ORDER_NUM: order of an NCC rule for the tetrahedron.
• TETRAHEDRON_NCC_RULE returns the points and weights of an NCC rule.
• TETRAHEDRON_NCC_RULE_NUM returns the number of NCC rules available.
• TETRAHEDRON_NCC_SUBORDER returns the suborders for an NCC rule.
• TETRAHEDRON_NCC_SUBORDER_NUM returns the number of suborders for an NCC rule.
• TETRAHEDRON_NCC_SUBRULE returns a compressed NCC rule.
• TETRAHEDRON_NCC_SUBRULE_01 returns a compressed NCC rule 1.
• TETRAHEDRON_NCC_SUBRULE_02 returns a compressed NCC rule 2.
• TETRAHEDRON_NCC_SUBRULE_03 returns a compressed NCC rule 3.
• TETRAHEDRON_NCC_SUBRULE_04 returns a compressed NCC rule 4.
• TETRAHEDRON_NCC_SUBRULE_05 returns a compressed NCC rule 5.
• TETRAHEDRON_NCC_SUBRULE_06 returns a compressed NCC rule 6.
• TETRAHEDRON_NCC_SUBRULE_07 returns a compressed NCC rule 7.
• TETRAHEDRON_VOLUME computes the volume of a tetrahedron.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.

Last revised on 14 June 2014.