NCC_TRIANGLE
Newton-Cotes Closed Quadrature Rules for the Triangle

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.

Licensing:

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

Languages:

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

Related Data and Programs:

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.

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:

• ncc_triangle.f90, the source code.
• ncc_triangle.sh, commands to compile the source code.

Examples and Tests:

• ncc_triangle_prb.f90, a sample calling program.
• ncc_triangle_prb.sh, commands to compile and run the sample program.
• ncc_triangle_prb_output.txt, the output file.

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.

• ncc_triangle_rule_01.png, a plot of rule 1.
• ncc_triangle_rule_02.png, a plot of rule 2.
• ncc_triangle_rule_03.png, a plot of rule 3.
• ncc_triangle_rule_04.png, a plot of rule 4.
• ncc_triangle_rule_05.png, a plot of rule 5.
• ncc_triangle_rule_06.png, a plot of rule 6.
• ncc_triangle_rule_07.png, a plot of rule 7.
• ncc_triangle_rule_08.png, a plot of rule 8.
• ncc_triangle_rule_09.png, a plot of rule 9.

List of Routines:

• FILE_NAME_INC increments a partially numeric filename.
• GET_UNIT returns a free FORTRAN unit number.
• 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.
• TIMESTRING writes the current YMDHMS date into a string.
• TRIANGLE_AREA computes the area of a triangle.
• TRIANGLE_POINTS_PLOT plots a triangle and some points.

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