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<html>
<head>
<title>
TRIANGLE_DUNAVANT_RULE - Quadrature Rules for the Triangle
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGLE_DUNAVANT_RULE <br> Quadrature Rules for the Triangle
</h1>
<hr>
<p>
<b>TRIANGLE_DUNAVANT_RULE</b>
is a MATLAB library which
defines 20 quadrature rules
over the interior of a triangle in 2D.
</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>TRIANGLE_DUNAVANT_RULE</b> is available in
<a href = "../../cpp_src/triangle_dunavant_rule/triangle_dunavant_rule.html">a C++ version</a> and
<a href = "../../f_src/triangle_dunavant_rule/triangle_dunavant_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangle_dunavant_rule/triangle_dunavant_rule.html">a MATLAB version</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/cube_felippa_rule/cube_felippa_rule.html">
CUBE_FELIPPA_RULE</a>,
a MATLAB library which
returns the points and weights of a Felippa quadrature rule
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../m_src/pyramid_felippa_rule/pyramid_felippa_rule.html">
PYRAMID_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../m_src/simplex_gm_rule/simplex_gm_rule.html">
SIMPLEX_GM_RULE</a>,
a MATLAB library which
defines Grundmann-Moeller quadrature rules
over the interior of a simplex in M dimensions.
</p>
<p>
<a href = "../../m_src/square_felippa_rule/square_felippa_rule.html">
SQUARE_FELIPPA_RULE</a>,
a MATLAB library which
returns the points and weights of a Felippa quadrature rule
over the interior of a square in 2D.
</p>
<p>
<a href = "../../m_src/stroud/stroud.html">
STROUD</a>,
a MATLAB library which
contains quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../m_src/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a MATLAB program which
investigates the polynomial exactness of a quadrature rule
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a MATLAB library which
defines Fekete rules for quadrature or interpolation
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_felippa_rule/triangle_felippa_rule.html">
TRIANGLE_FELIPPA_RULE</a>,
a MATLAB library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_lyness_rule/triangle_lyness_rule.html">
TRIANGLE_LYNESS_RULE</a>,
a MATLAB library which
returns Lyness-Jespersen quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a MATLAB program which
uses the Monte Carlo method to estimate integrals over a triangle.
</p>
<p>
<a href = "../../m_src/triangle_ncc_rule/triangle_ncc_rule.html">
TRIANGLE_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes Closed (NCC) quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_nco_rule/triangle_nco_rule.html">
TRIANGLE_NCO_RULE</a>,
a MATLAB library which
defines Newton-Cotes Open (NCO) quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_svg/triangle_svg.html">
TRIANGLE_SVG</a>,
a MATLAB library which
uses Scalable Vector Graphics (SVG) to plot a triangle and
any number of points, to illustrate quadrature rules and
sampling techniques.
</p>
<p>
<a href = "../../m_src/triangle_symq_rule/triangle_symq_rule.html">
TRIANGLE_SYMQ_RULE</a>,
a MATLAB library which
returns efficient symmetric quadrature rules,
with exactness up to total degree 50,
over the interior of an arbitrary triangle in 2D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../m_src/triangle_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a MATLAB library which
defines Wandzura rules for quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/wedge_felippa_rule/wedge_felippa_rule.html">
WEDGE_FELIPPA_RULE</a>,
a MATLAB library which
returns quadratures rules for approximating integrals
over the interior of the unit wedge in 3D.
</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_degree.m">dunavant_degree.m</a>
returns the degree of a Dunavant rule for the triangle.
</li>
<li>
<a href = "dunavant_order_num.m">dunavant_order_num.m</a>
returns the order of a given Dunavant rule for the triangle.
</li>
<li>
<a href = "dunavant_rule.m">dunavant_rule.m</a>
returns the points and weights of a Dunavant rule.
</li>
<li>
<a href = "dunavant_rule_num.m">dunavant_rule_num.m</a>
returns the number of available Dunavant rules.
</li>
<li>
<a href = "dunavant_suborder.m">dunavant_suborder.m</a>
returns the suborders for a Dunavant rule.
</li>
<li>
<a href = "dunavant_suborder_num.m">dunavant_suborder_num.m</a>
returns the number of suborders for a Dunavant rule.
</li>
<li>
<a href = "dunavant_subrule.m">dunavant_subrule.m</a>
returns a compressed Dunavant rule.
</li>
<li>
<a href = "i4_modp.m">i4_modp.m</a>
returns the nonnegative remainder of integer division.
</li>
<li>
<a href = "i4_wrap.m">i4_wrap.m</a>
forces an integer to lie between given limits by wrapping.
</li>
<li>
<a href = "reference_to_physical_t3.m">reference_to_physical_t3.m</a>
maps T3 reference points to physical points.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints the current YMDHMS date as a time stamp.
</li>
<li>
<a href = "triangle_area.m">triangle_area.m</a>
computes the area of a triangle.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "triangle_dunavant_rule_test.m">triangle_dunavant_rule_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "triangle_dunavant_rule_test01.m">triangle_dunavant_rule_test01.m</a>,
tests DUNAVANT_ORDER_NUM.
</li>
<li>
<a href = "triangle_dunavant_rule_test02.m">triangle_dunavant_rule_test02.m</a>,
tests DUNAVANT_RULE by summing the weights.
</li>
<li>
<a href = "triangle_dunavant_rule_test03.m">triangle_dunavant_rule_test03.m</a>,
tests DUNAVANT_RULE by summing the coordinates.
</li>
<li>
<a href = "triangle_dunavant_rule_test04.m">triangle_dunavant_rule_test04.m</a>,
tests DUNAVANT_ORDER by integrating monomials in the
unit triangle.
</li>
<li>
<a href = "triangle_dunavant_rule_test05.m">triangle_dunavant_rule_test05.m</a>,
tests REFERENCE_TO_PHYSICAL_T3 by transforming a rule
from the unit triangle to another triangle.
</li>
<li>
<a href = "triangle_dunavant_rule_test_output.txt">triangle_dunavant_rule_test_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 29 June 2014.
</i>
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