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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 C++ library which
defines the weights and abscisass for 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 = "../../cpp_src/cube_felippa_rule/cube_felippa_rule.html">
CUBE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a cube in 3D.
</p>
<p>
<a href = "../../cpp_src/pyramid_felippa_rule/pyramid_felippa_rule.html">
PYRAMID_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a pyramid in 3D.
</p>
<p>
<a href = "../../cpp_src/simplex_gm_rule/simplex_gm_rule.html">
SIMPLEX_GM_RULE</a>,
a C++ library which
defines Grundmann-Moeller quadrature rules
over the interior of the simplex in M dimensions.
</p>
<p>
<a href = "../../cpp_src/square_felippa_rule/square_felippa_rule.html">
SQUARE_FELIPPA_RULE</a>,
a C++ library which
returns the points and weights of a Felippa quadrature rule
over the interior of a square in 2D.
</p>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ library which
defines quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_felippa_rule/tetrahedron_felippa_rule.html">
TETRAHEDRON_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a C++ library which
defines Fekete rules for interpolation or quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_felippa_rule/triangle_felippa_rule.html">
TRIANGLE_FELIPPA_RULE</a>,
a C++ library which
returns Felippa's quadratures rules for approximating integrals
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_lyness_rule/triangle_lyness_rule.html">
TRIANGLE_LYNESS_RULE</a>,
a C++ library which
returns Lyness-Jespersen quadrature rules for the triangle.
</p>
<p>
<a href = "../../cpp_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a C++ program which
uses the Monte Carlo method to estimate integrals over a triangle.
</p>
<p>
<a href = "../../cpp_src/triangle_ncc_rule/triangle_ncc_rule.html">
TRIANGLE_NCC_RULE</a>,
a C++ library which
defines Newton-Cotes Closed (NCC) quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_nco_rule/triangle_nco_rule.html">
TRIANGLE_NCO_RULE</a>,
a C++ library which
defines Newton-Cotes Open (NCO) quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_symq_rule/triangle_symq_rule.html">
TRIANGLE_SYMQ_RULE</a>,
a C++ 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 = "../../cpp_src/triangle_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a C++ library which
defines Wandzura rules for quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/wedge_felippa_rule/wedge_felippa_rule.html">
WEDGE_FELIPPA_RULE</a>,
a C++ 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 = "triangle_dunavant_rule.cpp">triangle_dunavant_rule.cpp</a>, the source code.
</li>
<li>
<a href = "triangle_dunavant_rule.hpp">triangle_dunavant_rule.hpp</a>, the include file.
</li>
<li>
<a href = "triangle_dunavant_rule.sh">triangle_dunavant_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "triangle_dunavant_rule_prb.cpp">triangle_dunavant_rule_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "triangle_dunavant_rule_prb.sh">triangle_dunavant_rule_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "triangle_dunavant_rule_prb_output.txt">triangle_dunavant_rule_prb_output.txt</a>,
the output ffile.
</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>DUNAVANT_SUBRULE_01</b> returns a compressed Dunavant rule 1.
</li>
<li>
<b>DUNAVANT_SUBRULE_02</b> returns a compressed Dunavant rule 2.
</li>
<li>
<b>DUNAVANT_SUBRULE_03</b> returns a compressed Dunavant rule 3.
</li>
<li>
<b>DUNAVANT_SUBRULE_04</b> returns a compressed Dunavant rule 4.
</li>
<li>
<b>DUNAVANT_SUBRULE_05</b> returns a compressed Dunavant rule 5.
</li>
<li>
<b>DUNAVANT_SUBRULE_06</b> returns a compressed Dunavant rule 6.
</li>
<li>
<b>DUNAVANT_SUBRULE_07</b> returns a compressed Dunavant rule 7.
</li>
<li>
<b>DUNAVANT_SUBRULE_08</b> returns a compressed Dunavant rule 8.
</li>
<li>
<b>DUNAVANT_SUBRULE_09</b> returns a compressed Dunavant rule 9.
</li>
<li>
<b>DUNAVANT_SUBRULE_10</b> returns a compressed Dunavant rule 10.
</li>
<li>
<b>DUNAVANT_SUBRULE_11</b> returns a compressed Dunavant rule 11.
</li>
<li>
<b>DUNAVANT_SUBRULE_12</b> returns a compressed Dunavant rule 12.
</li>
<li>
<b>DUNAVANT_SUBRULE_13</b> returns a compressed Dunavant rule 13.
</li>
<li>
<b>DUNAVANT_SUBRULE_14</b> returns a compressed Dunavant rule 14.
</li>
<li>
<b>DUNAVANT_SUBRULE_15</b> returns a compressed Dunavant rule 15.
</li>
<li>
<b>DUNAVANT_SUBRULE_16</b> returns a compressed Dunavant rule 16.
</li>
<li>
<b>DUNAVANT_SUBRULE_17</b> returns a compressed Dunavant rule 17.
</li>
<li>
<b>DUNAVANT_SUBRULE_18</b> returns a compressed Dunavant rule 18.
</li>
<li>
<b>DUNAVANT_SUBRULE_19</b> returns a compressed Dunavant rule 19.
</li>
<li>
<b>DUNAVANT_SUBRULE_20</b> returns a compressed Dunavant rule 20.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two integers.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two integers.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of integer division.
</li>
<li>
<b>I4_WRAP</b> forces an integer to lie between given limits by wrapping.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to an R8.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_T3</b> maps T3 reference points to physical points.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</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>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 29 June 2014.
</i>
<!-- John Burkardt -->
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