TRIANGLE_NCC_RULE
Newton-Cotes Closed (NCC) Quadrature for the Triangle

TRIANGLE_NCC_RULE is a FORTRAN90 library which defines Newton Cotes closed (NCC) 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 "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.

Licensing:

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

Languages:

TRIANGLE_NCC_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 library which returns the points and weights of a Felippa quadrature rule over the interior of a cube in 3D.

LINE_NCC_RULE, a FORTRAN90 library which computes a Newton Cotes Closed (NCC) quadrature rule for the line, that is, for an interval of the form [A,B], using equally spaced points which include the endpoints.

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

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

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

STROUD, a FORTRAN90 library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

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

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

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

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

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

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

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

TRIANGLE_MONTE_CARLO, a FORTRAN90 program which uses the Monte Carlo method to estimate integrals over the interior of the triangle in 2D.

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

TRIANGLE_SVG, a FORTRAN90 library which uses Scalable Vector Graphics (SVG) to plot a triangle and any number of points, to illustrate quadrature rules and sampling techniques.

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

WEDGE_FELIPPA_RULE, a FORTRAN90 library which returns quadratures rules for approximating integrals over the interior of the unit wedge in 3D.

Reference:

1. Gisela Engeln-Muellges, Frank Uhlig,
   Numerical Algorithms with C,
   Springer, 1996,
   ISBN: 3-540-60530-4,
   LC: QA297.E56213.
2. Peter Silvester,
   Symmetric Quadrature Formulae for Simplexes,
   Mathematics of Computation,
   Volume 24, Number 109, January 1970, pages 95-100.

Source Code:

• triangle_ncc_rule.f90, the source code.

Examples and Tests:

• triangle_ncc_rule_test.f90, a sample calling program.
• triangle_ncc_rule_test.txt, the output file.

List of Routines:

• I4_MODP returns the nonnegative remainder of integer division.
• I4_WRAP forces an integer to lie between given limits by wrapping.
• NCC_TRIANGLE_DEGREE returns the degree of an NCC rule for the triangle.
• NCC_TRIANGLE_ORDER_NUM returns the order of an NCC rule for the triangle.
• NCC_TRIANGLE_RULE returns the points and weights of an NCC rule.
• NCC_TRIANGLE_RULE_NUM returns the number of Newton-Cotes closed rules available.
• NCC_TRIANGLE_SUBORDER returns the suborders for an NCC rule.
• NCC_TRIANGLE_SUBORDER_NUM returns the number of suborders for an NCC rule.
• NCC_TRIANGLE_SUBRULE returns a compressed NCC rule.
• REFERENCE_TO_PHYSICAL_T3 maps T3 reference points to physical points.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TRIANGLE_AREA computes the area of a triangle.

You can go up one level to the FORTRAN90 source codes.