WANDZURA
High Order Quadrature in a Triangle.

WANDZURA is a FORTRAN90 libary, using double precision arithmetic, which can return any of six Wandzura rules for high order quadrature in a triangle.

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

Licensing:

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

Related Data and Programs:

CLENSHAW_CURTIS is a FORTRAN90 library which sets a Clenshaw Curtis quadrature grid in multiple dimensions.

DUNAVANT is a FORTRAN90 library which defines Dunavant rules for quadrature on a triangle.

FEKETE is a FORTRAN90 library which defines Fekete rules for interpolation or quadrature on a triangle.

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

FEM_SAMPLE is a FORTRAN90 library of routines for evaluating a finite element function defined on an order 3 or order 6 triangulation.

FEMPACK is a FORTRAN90 library of routines for finite element calculations.

GM_RULES is a FORTRAN90 library which defines Grundmann-Moeller rules for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

NCC_TRIANGLE is a FORTRAN90 library which defines Newton-Cotes closed quadrature rules on a triangle.

NCO_TRIANGLE is a FORTRAN90 library which defines Newton-Cotes open quadrature rules on a triangle.

NINT_EXACTNESS_TRI is a FORTRAN90 program which investigates the polynomial exactness of a quadrature rule for the triangle.

NINTLIB is a FORTRAN90 library containing a variety of routines for numerical estimation of integrals in multiple dimensions.

QUADRATURE_RULES_TRI is a dataset directory of triples of files defining various quadrature rules on triangles.

QUADRULE is a FORTRAN90 library for defining quadrature rules on a variety of intervals with different weight functions.

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

TEST_TRI_INT is a FORTRAN90 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 MATLAB 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.

Source Code:

• wandzura.f90, the source code.
• wandzura.csh, commands to compile the source code.

Examples and Tests:

• wandzura_prb.f90, a sample calling program.
• wandzura_prb.csh, commands to compile and run the sample program.
• wandzura_prb_output.txt, the output from a run of the sample program.

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.

• 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.

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 I4 division.
• I4_WRAP forces an I4 to lie between given limits by wrapping.
• 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.
• WANDZURA_DEGREE returns the degree of a given Wandzura rule for the triangle.
• WANDZURA_ORDER_NUM returns the order of a Wandzura rule for the triangle.
• WANDZURA_RULE returns the points and weights of a Wandzura rule.
• WANDZURA_RULE_NUM returns the number of Wandzura rules available.
• WANDZURA_SUBORDER returns the suborders for a Wandzura rule.
• WANDZURA_SUBORDER_NUM returns the number of suborders for a Wandzura rule.
• WANDZURA_SUBRULE returns a compressed Wandzura rule.
• WANDZURA_SUBRULE2 returns a compressed Wandzura rule.

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