WANDZURA
Quadrature Rules for Triangles

WANDZURA is a MATLAB library, using double precision arithmetic, which can return any of six Wandzura rules for quadrature in a triangle.

There are six rules, which have polynomial degree of exactness of 5, 10, 15, 20, 25, and 30.

Related Data and Programs:

CLENSHAW_CURTIS is a MATLAB library which can set up a Clenshaw Curtis quadrature grid in multiple dimensions.

DUNAVANT is a MATLAB library which can set up a Dunavant rule for quadrature in a triangle.

FEKETE is a MATLAB library which can set up a Fekete rule for interpolation or quadrature in a triangle.

FEM_SAMPLE is a MATLAB library of routines for evaluating a finite element function defined on an order 3 or order 6 triangulation.

FEMPACK is a MATLAB library of routines for finite element calculations.

GM_RULES is a MATLAB library of routines for defining a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

NCC_TRIANGLE is a MATLAB library defining Newton-Cotes closed quadrature rules on a triangle.

NCO_TRIANGLE is a MATLAB library defining Newton-Cotes open quadrature rules on a triangle.

NINT_EXACTNESS_TRI is an executable MATLAB program which investigates the polynomial exactness of a quadrature rule for the triangle.

NINTLIB is a MATLAB library containing a variety of routines for numerical estimation of integrals in multiple dimensions.

QUADRULE is a MATLAB library for defining quadrature rules on a variety of intervals with different weight functions.

STROUD is a MATLAB library containing quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TEST_TRI_INT is a MATLAB library of functions that can be used to test algorithms for quadrature over a triangle.

TOMS612 is a FORTRAN77 library of routines which can estimate the integral of a function over a triangle.

TOMS706 is a FORTRAN77 library which estimates the integral of a function over a triangulated region.

WANDZURA is also available in a C++ version and a FORTRAN90 version.

Reference:

1. James Lyness, Dennis Jespersen,
   Moderate Degree Symmetric Quadrature Rules for the Triangle,
   Journal of the Institute of Mathematics and its Applications,
   Volume 15, Number 1, February 1975, pages 19-32.
2. Stephen Wandzura, Hong Xiao,
   Symmetric Quadrature Rules on a Triangle,
   Computers and Mathematics with Applications,
   Volume 45, Number 12, June 2003, pages 1829-1840.

Tar File:

A GZIP'ed TAR file of the contents of this directory is available. This is only done as a convenience for users who want ALL the files, and don't want to download them individually. This is not a convenience for me, so don't be surprised if the tar file is somewhat out of date.

• wandzura_m_src.tar.csh the commands used to create the GZIP'ed TAR file.
• wandzura_m_src.tar.gz the GZIP'ed TAR file.

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.
• reference_to_physical_t3.m maps T3 reference points to physical points.
• timestamp.m prints the current YMDHMS date as a time stamp.
• timestring.m writes the current YMDHMS date into a string.
• triangle_area.m computes the area of a triangle.
• triangle_points_plot.m plots a triangle and some points.
• wandzura_degree.m returns the degree of a given Wandzura rule for the triangle.
• wandzura_order_num.m returns the order of a given Wandzura rule for the triangle.
• wandzura_rule.m returns the points and weights of a Wandzura rule.
• wandzura_rule_num.m returns the number of Wandzura rules available.
• wandzura_suborder.m returns the suborders for a Wandzura rule.
• wandzura_suborder_num.m returns the number of suborders for a Wandzura rule.
• wandzura_subrule.m returns a compressed Wandzura rule.
• wandzura_subrule2.m returns a compressed Wandzura rule, using the nonstandard reference triangle.

Examples and Tests:

• wandzura_test.m, runs all the tests.
• wandzura_test01.m, tests WANDZURA_RULE_NUM, WANDZURA_DEGREE, and WANDZURA_ORDER_NUM.
• wandzura_test02.m, tests WANDZURA_RULE by summing the weights.
• wandzura_test03.m, tests WANDZURA_RULE by summing the barycentric coordinates.
• wandzura_test04.m, tests WANDZURA_ORDER by integrating monomials in the unit triangle.
• wandzura_test05.m, tests WANDZURA_RULE by plotting the points.
• wandzura_test06.m, tests REFERENCE_TO_PHYSICAL_T3 by transforming a Wandzura rule from the unit triangle to another triangle.
• wandzura_test.out, the output from a run of the sample program.

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.

• wandzura_rule_1.png, a plot of rule 1.
• wandzura_rule_2.png, a plot of rule 2.
• wandzura_rule_3.png, a plot of rule 3.
• wandzura_rule_4.png, a plot of rule 4.
• wandzura_rule_5.png, a plot of rule 5.
• wandzura_rule_6.png, a plot of rule 6.

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