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<html>
<head>
<title>
WANDZURA - High Order Quadrature in a Triangle.
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
WANDZURA <br> High Order Quadrature in a Triangle.
</h1>
<hr>
<p>
<b>WANDZURA</b>
is a FORTRAN90 library which
can return any of six Wandzura
rules for high order quadrature in a triangle.
</p>
<p>
There are six rules, which have polynomial degree of exactness
of 5, 10, 15, 20, 25, and 30.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>WANDZURA</b> is available in
<a href = "../../cpp_src/wandzura/wandzura.html">a C++ version</a> and
<a href = "../../f_src/wandzura/wandzura.html">a FORTRAN90 version</a> and
<a href = "../../m_src/wandzura/wandzura.html">a MATLAB version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/dunavant/dunavant.html">
DUNAVANT</a>,
a FORTRAN90 library which
defines Dunavant rules for quadrature on a triangle.
</p>
<p>
<a href = "../../f_src/fekete/fekete.html">
FEKETE</a>,
a FORTRAN90 library which
defines Fekete rules for interpolation or quadrature on a triangle.
</p>
<p>
<a href = "../../f_src/felippa/felippa.html">
FELIPPA</a>,
a FORTRAN90 library which
defines quadrature rules for lines, triangles, quadrilaterals,
pyramids, wedges, tetrahedrons and hexahedrons.
</p>
<p>
<a href = "../../f_src/gm_rule/gm_rule.html">
GM_RULE</a>,
a FORTRAN90 library which
defines Grundmann-Moeller rules for quadrature over a triangle, tetrahedron,
or general M-dimensional simplex.
</p>
<p>
<a href = "../../f_src/lyness_rule/lyness_rule.html">
LYNESS_RULE</a>,
a FORTRAN90 library which
returns Lyness-Jespersen quadrature rules for the triangle.
</p>
<p>
<a href = "../../f_src/ncc_triangle/ncc_triangle.html">
NCC_TRIANGLE</a>,
a FORTRAN90 library which
defines Newton-Cotes closed quadrature rules on a triangle.
</p>
<p>
<a href = "../../f_src/nco_triangle/nco_triangle.html">
NCO_TRIANGLE</a>,
a FORTRAN90 library which
defines Newton-Cotes open quadrature rules on a triangle.
</p>
<p>
<a href = "../../f_src/nintlib/nintlib.html">
NINTLIB</a>,
a FORTRAN90 library which
contains routines for numerical estimation of integrals in M dimensions.
</p>
<p>
<a href = "../../datasets/quadrature_rules_tri/quadrature_rules_tri.html">
QUADRATURE_RULES_TRI</a>,
a dataset directory of triples of files which
defines various quadrature
rules on triangles.
</p>
<p>
<a href = "../../f_src/quadrule/quadrule.html">
QUADRULE</a>,
a FORTRAN90 library which
defines quadrature rules on a
variety of intervals with different weight functions.
</p>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and M dimensions.
</p>
<p>
<a href = "../../f_src/test_tri_int/test_tri_int.html">
TEST_TRI_INT</a>,
a FORTRAN90 library which
tests algorithms
for quadrature over a triangle.
</p>
<p>
<a href = "../../f77_src/toms612/toms612.html">
TOMS612</a>,
a FORTRAN77 library which
estimates the integral of a function over a triangle.
</p>
<p>
<a href = "../../f77_src/toms706/toms706.html">
TOMS706</a>,
a FORTRAN77 library which
estimates the integral of a function
over a triangulated region.
</p>
<p>
<a href = "../../f_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a FORTRAN90 program which
investigates the polynomial exactness of a quadrature rule for the triangle.
</p>
<p>
<a href = "../../f_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a FORTRAN90 program which
uses the Monte Carlo method to estimate integrals over a triangle.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
James Lyness, Dennis Jespersen,<br>
Moderate Degree Symmetric Quadrature Rules for the Triangle,<br>
Journal of the Institute of Mathematics and its Applications,<br>
Volume 15, Number 1, February 1975, pages 19-32.
</li>
<li>
Stephen Wandzura, Hong Xiao,<br>
Symmetric Quadrature Rules on a Triangle,<br>
Computers and Mathematics with Applications,<br>
Volume 45, Number 12, June 2003, pages 1829-1840.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "wandzura.f90">wandzura.f90</a>, the source code.
</li>
<li>
<a href = "wandzura.sh">wandzura.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "wandzura_prb.f90">wandzura_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "wandzura_prb.sh">wandzura_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "wandzura_prb_output.txt">wandzura_prb_output.txt</a>,
the output from a run of the sample program.
</li>
</ul>
</p>
<p>
One of the tests in the sample calling program creates
<a href = "../../data/eps/eps.html">EPS</a> files of
the points in the unit triangle. These have been converted
to <a href = "../../data/png/png.html">PNG</a> files for
display here.
<ul>
<li>
<a href = "wandzura_rule_1.png">wandzura_rule_1.png</a>,
a plot of rule 1.
</li>
<li>
<a href = "wandzura_rule_2.png">wandzura_rule_2.png</a>,
a plot of rule 2.
</li>
<li>
<a href = "wandzura_rule_3.png">wandzura_rule_3.png</a>,
a plot of rule 3.
</li>
<li>
<a href = "wandzura_rule_4.png">wandzura_rule_4.png</a>,
a plot of rule 4.
</li>
<li>
<a href = "wandzura_rule_5.png">wandzura_rule_5.png</a>,
a plot of rule 5.
</li>
<li>
<a href = "wandzura_rule_6.png">wandzura_rule_6.png</a>,
a plot of rule 6.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FILE_NAME_INC</b> increments a partially numeric filename.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of I4 division.
</li>
<li>
<b>I4_WRAP</b> forces an I4 to lie between given limits by wrapping.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_T3</b> maps T3 reference points to physical points.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRIANGLE_AREA</b> computes the area of a triangle.
</li>
<li>
<b>TRIANGLE_POINTS_PLOT</b> plots a triangle and some points.
</li>
<li>
<b>WANDZURA_DEGREE</b> returns the degree of a given Wandzura rule for the triangle.
</li>
<li>
<b>WANDZURA_ORDER_NUM</b> returns the order of a Wandzura rule for the triangle.
</li>
<li>
<b>WANDZURA_RULE</b> returns the points and weights of a Wandzura rule.
</li>
<li>
<b>WANDZURA_RULE_NUM</b> returns the number of Wandzura rules available.
</li>
<li>
<b>WANDZURA_SUBORDER</b> returns the suborders for a Wandzura rule.
</li>
<li>
<b>WANDZURA_SUBORDER_NUM</b> returns the number of suborders for a Wandzura rule.
</li>
<li>
<b>WANDZURA_SUBRULE</b> returns a compressed Wandzura rule.
</li>
<li>
<b>WANDZURA_SUBRULE2</b> returns a compressed Wandzura rule.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 01 January 2010.
</i>
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