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<html>
<head>
<title>
DUNAVANT - Quadrature Rules for the Triangle
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
DUNAVANT <br> Quadrature Rules for the Triangle
</h1>
<hr>
<p>
<b>DUNAVANT</b>
is a FORTRAN90 library which
defines the weights and abscisass for a sequence of
20 quadrature rules on a triangle, which are exact for polynomials
up to degree 20.
</p>
<p>
These rules are almost optimal, in the sense that, for each polynomial
degree, the number of points used in the rule is close to, or equal to,
the theoretical minimum possible value.
</p>
<p>
A few of the rules include one or two points which are "slightly"
outside the triangle; a few of the rules include weights which
are negative. Both of these occurrences are generally undesirable.
</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>DUNAVANT</b> is available in
<a href = "../../cpp_src/dunavant/dunavant.html">a C++ version</a> and
<a href = "../../f_src/dunavant/dunavant.html">a FORTRAN90 version</a> and
<a href = "../../m_src/dunavant/dunavant.html">a MATLAB version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/fekete/fekete.html">
FEKETE</a>,
a FORTRAN90 library which
defines Fekete rules for quadrature or interpolation 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 a Grundmann-Moeller
rule 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 a variety
of routines for numerical estimation of integrals in multiple dimensions.
</p>
<p>
<a href = "../../datasets/quadrature_rules_tri/quadrature_rules_tri.html">
QUADRATURE_RULES_TRI</a>,
a dataset directory which
contains 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
can be used to test 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>
<p>
<a href = "../../f_src/wandzura/wandzura.html">
WANDZURA</a>,
a FORTRAN90 library which
defines Wandzura rules for quadrature on a triangle.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
David Dunavant,<br>
High Degree Efficient Symmetrical Gaussian Quadrature Rules
for the Triangle,<br>
International Journal for Numerical Methods in Engineering,<br>
Volume 21, 1985, pages 1129-1148.
</li>
<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>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "dunavant.f90">dunavant.f90</a>, the source code.
</li>
<li>
<a href = "dunavant.sh">dunavant.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "dunavant_prb.f90">dunavant_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "dunavant_prb.sh">dunavant_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "dunavant_prb_output.txt">dunavant_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 Dunavant 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 = "dunavant_rule_01.png">dunavant_rule_01.png</a>,
a plot of rule 1.
</li>
<li>
<a href = "dunavant_rule_02.png">dunavant_rule_02.png</a>,
a plot of rule 2.
</li>
<li>
<a href = "dunavant_rule_03.png">dunavant_rule_03.png</a>,
a plot of rule 3.
</li>
<li>
<a href = "dunavant_rule_04.png">dunavant_rule_04.png</a>,
a plot of rule 4.
</li>
<li>
<a href = "dunavant_rule_05.png">dunavant_rule_05.png</a>,
a plot of rule 5.
</li>
<li>
<a href = "dunavant_rule_06.png">dunavant_rule_06.png</a>,
a plot of rule 6.
</li>
<li>
<a href = "dunavant_rule_07.png">dunavant_rule_07.png</a>,
a plot of rule 7.
</li>
<li>
<a href = "dunavant_rule_08.png">dunavant_rule_08.png</a>,
a plot of rule 8.
</li>
<li>
<a href = "dunavant_rule_09.png">dunavant_rule_09.png</a>,
a plot of rule 9.
</li>
<li>
<a href = "dunavant_rule_10.png">dunavant_rule_10.png</a>,
a plot of rule 10.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>DUNAVANT_DEGREE</b> returns the degree of a Dunavant rule for the triangle.
</li>
<li>
<b>DUNAVANT_ORDER_NUM</b> returns the order of a Dunavant rule for the triangle.
</li>
<li>
<b>DUNAVANT_RULE</b> returns the points and weights of a Dunavant rule.
</li>
<li>
<b>DUNAVANT_RULE_NUM</b> returns the number of Dunavant rules available.
</li>
<li>
<b>DUNAVANT_SUBORDER</b> returns the suborders for a Dunavant rule.
</li>
<li>
<b>DUNAVANT_SUBORDER_NUM</b> returns the number of suborders for a Dunavant rule.
</li>
<li>
<b>DUNAVANT_SUBRULE</b> returns a compressed Dunavant rule.
</li>
<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>
</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 27 December 2010.
</i>
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