NCO_TRIANGLE
Newton-Cotes Open Quadrature Rules for the Triangle

NCO_TRIANGLE is a MATLAB library which defines the weights and abscisass for a sequence of 9 Newton-Cotes open 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 "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 GNU LGPL license.

Languages:

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

Related Data and Programs:

DUNAVANT, a MATLAB library which defines Dunavant rules for quadrature on a triangle.

FEKETE, a MATLAB library which defines Fekete rules for quadrature or interpolation on a triangle.

FELIPPA, a MATLAB library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

GM_RULE, a MATLAB library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

LYNESS_RULE, a MATLAB library which returns Lyness-Jespersen quadrature rules for the triangle.

NCC_TETRAHEDRON, a MATLAB library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

NCC_TRIANGLE, a MATLAB library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TETRAHEDRON, a MATLAB library which defines Newton-Cotes open quadrature rules on a tetrahedron.

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

TEST_TRI_INT, a MATLAB 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.

TRIANGLE_MONTE_CARLO, a MATLAB program which uses the Monte Carlo method to estimate integrals over a triangle.

WANDZURA, a MATLAB library which defines Dunavant rules for quadrature on a triangle.

Reference:

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

Source Code:

• file_name_inc.m increments a partially numeric filename.
• i4_modp.m returns the nonnegative remainder of integer division.
• i4_wrap.m forces an integer to lie between given limits by wrapping.
• nco_triangle_degree.m returns the degree of a given NCO rule for the triangle.
• nco_triangle_order_num.m returns the order of a given NCO rule for the triangle.
• nco_triangle_rule.m returns the points and weights of an NCO rule.
• nco_triangle_rule_num.m returns the number of NCO rules available.
• nco_triangle_suborder.m returns the suborders for an NCO rule.
• nco_triangle_suborder_num.m returns the number of suborders for an NCO rule.
• nco_triangle_subrule.m returns a compressed NCO rule.
• reference_to_physical_t3.m maps T3 reference points to physical points.
• timestamp.m prints the current YMDHMS date as a time stamp.
• triangle_area.m computes the area of a triangle.
• triangle_points_plot.m plots a triangle and some points.

Examples and Tests:

• nco_triangle_test.m, runs all the tests.
• nco_triangle_test01.m, tests NCO_TRIANGLE_RULE_NUM, NCO_TRIANGLE_DEGREE, and NCO_TRIANGLE_ORDER_NUM.
• nco_triangle_test02.m, tests NCO_TRIANGLE_RULE by summing the weights.
• nco_triangle_test03.m, tests NCO_TRIANGLE_RULE by summing the barycentric coordinates.
• nco_triangle_test04.m, tests NCO_TRIANGLE_ORDER by integrating monomials in the unit triangle.
• nco_triangle_test05.m, tests NCO_TRIANGLE_RULE by plotting the points.
• nco_triangle_test06.m, tests REFERENCE_TO_PHYSICAL_T3 by transforming an NCC rule from the unit triangle to another triangle.
• nco_triangle_test_output.txt, the output file.

One of the tests in the sample calling program creates EPS files of the points in the unit triangle. These have been converted to PNG files for display here.

• nco_triangle_rule_1.png, a plot of rule 1.
• nco_triangle_rule_2.png, a plot of rule 2.
• nco_triangle_rule_3.png, a plot of rule 3.
• nco_triangle_rule_4.png, a plot of rule 4.
• nco_triangle_rule_5.png, a plot of rule 5.
• nco_triangle_rule_6.png, a plot of rule 6.
• nco_triangle_rule_7.png, a plot of rule 7
• nco_triangle_rule_8.png, a plot of rule 8
• nco_triangle_rule_9.png, a plot of rule 9

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