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<html>
<head>
<title>
TRIANGLE_SYMQ_RULE - Symmetric Quadrature Rules for Triangles.
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGLE_SYMQ_RULE <br> Symmetric Quadrature Rules for Triangles.
</h1>
<hr>
<p>
<b>TRIANGLE_SYMQ_RULE</b>
is a C++ library which
returns 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>
The original source code, from which this library was developed,
is available from the Courant Mathematics and Computing Laboratory, at
<a href = "http://www.cims.nyu.edu/cmcl/quadratures/quadratures.html">
http://www.cims.nyu.edu/cmcl/quadratures/quadratures.html </a>,
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files 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_SYMQ_RULE</b> is available in
<a href = "../../c_src/triangle_symq_rule/triangle_symq_rule.html">a C version</a> and
<a href = "../../cpp_src/triangle_symq_rule/triangle_symq_rule.html">a C++ version</a> and
<a href = "../../f77_src/triangle_symq_rule/triangle_symq_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/triangle_symq_rule/triangle_symq_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangle_symq_rule/triangle_symq_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/gnuplot/gnuplot.html">
GNUPLOT</a>,
C++ programs which
illustrate how a program can write data and command files
so that gnuplot can create plots of the program results.
</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 a 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/square_symq_rule/square_symq_rule.html">
SQUARE_SYMQ_RULE</a>,
a C++ library which
returns symmetric quadrature rules,
with exactness up to total degree 20,
over the interior of the symmetric square in 2D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../cpp_src/stroud/stroud.html">
STROUD</a>,
a C++ library which
defines quadrature rules for a variety of M-dimensional regions,
including the interior of the square, cube and hypercube, the pyramid,
cone and ellipse, the hexagon, the M-dimensional octahedron,
the circle, sphere and hypersphere, the triangle, tetrahedron and simplex,
and the surface of the circle, sphere and hypersphere.
</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/toms886/toms886.html">
TOMS886</a>,
a C++ library which
defines the Padua points for interpolation in a 2D region,
including the rectangle, triangle, and ellipse,
by Marco Caliari, Stefano de Marchi, Marco Vianello.
This is a version of ACM TOMS algorithm 886.
</p>
<p>
<a href = "../../cpp_src/triangle_dunavant_rule/triangle_dunavant_rule.html">
TRIANGLE_DUNAVANT_RULE</a>,
a C++ library which
defines Dunavant rules for quadrature
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_integrals/triangle_integrals.html">
TRIANGLE_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit 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
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a C++ library which
uses the Monte Carlo method to estimate the integral of a function
over the interior of the unit triangle in 2D.
</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_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a C++ library which
returns quadrature rules of exactness 5, 10, 15, 20, 25 and 30
over the interior of the 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>
Hong Xiao, Zydrunas Gimbutas, <br>
A numerical algorithm for the construction of efficient quadrature
rules in two and higher dimensions,<br>
Computers and Mathematics with Applications,<br>
Volume 59, 2010, pages 663-676.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "triangle_symq_rule.cpp">triangle_symq_rule.cpp</a>, the source code.
</li>
<li>
<a href = "triangle_symq_rule.hpp">triangle_symq_rule.hpp</a>, the include file.
</li>
<li>
<a href = "triangle_symq_rule.sh">triangle_symq_rule.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "triangle_symq_rule_prb.cpp">triangle_symq_rule_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "triangle_symq_rule_prb.sh">triangle_symq_rule_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "triangle_symq_rule_prb_output.txt">
triangle_symq_rule_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
EQUI08 is a degree 8 rule in an equilateral triangle.
<ul>
<li>
<a href = "equi08.txt">equi08.txt</a>,
the node coordinates and weights.
</li>
<li>
<a href = "equi08_commands.txt">equi08_commands.txt</a>,
gnuplot commands to create a plot.
</li>
<li>
<a href = "equi08_nodes.txt">equi08_nodes.txt</a>,
the node coordinates.
</li>
<li>
<a href = "equi08_vertices.txt">equi08_vertices.txt</a>,
the triangle vertices.
</li>
<li>
<a href = "equi08.png">equi08.png</a>,
the PNG image of the point locations.
</li>
</ul>
</p>
<p>
SIMP08 is a degree 8 rule in a simplex.
<ul>
<li>
<a href = "simp08.txt">simp08.txt</a>,
the node coordinates and weights.
</li>
<li>
<a href = "simp08_commands.txt">simp08_commands.txt</a>,
gnuplot commands to create a plot.
</li>
<li>
<a href = "simp08_nodes.txt">simp08_nodes.txt</a>,
the node coordinates.
</li>
<li>
<a href = "simp08_vertices.txt">simp08_vertices.txt</a>,
the triangle vertices.
</li>
<li>
<a href = "simp08.png">simp08.png</a>,
the PNG image of the point locations.
</li>
</ul>
</p>
<p>
USER08 is a degree 8 rule in a user specified triangle at (1,0),
(4,4), (0,3).
<ul>
<li>
<a href = "user08.txt">user08.txt</a>,
the node coordinates and weights.
</li>
<li>
<a href = "user08_commands.txt">user08_commands.txt</a>,
gnuplot commands to create a plot.
</li>
<li>
<a href = "user08_nodes.txt">user08_nodes.txt</a>,
the node coordinates.
</li>
<li>
<a href = "user08_vertices.txt">user08_vertices.txt</a>,
the triangle vertices.
</li>
<li>
<a href = "user08.png">simp08.png</a>,
the PNG image of the point locations.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>KJACOPOLS2</b> evaluates Jacobi polynomials and derivatives.
</li>
<li>
<b>KJACOPOLS</b> evaluates Jacobi polynomials.
</li>
<li>
<b>KLEGEYPOLS3</b> evaluate scaled Legendre polynomials and derivatives.
</li>
<li>
<b>KLEGEYPOLS</b> evaluates scaled Legendre polynomials.
</li>
<li>
<b>ORTHO2EVA0</b> evaluates the orthonormal polynomials on the triangle.
</li>
<li>
<b>ORTHO2EVA30:</b> orthonormal polynomials and derivatives on triangle.
</li>
<li>
<b>ORTHO2EVA3:</b> orthogonal polynomial values and derivatives, reference triangle.
</li>
<li>
<b>ORTHO2EVA</b> evaluates orthogonal polynomials on the reference triangle.
</li>
<li>
<b>QUAECOPY2</b> copies a quadrature rule into a user arrays X, Y, and W.
</li>
<li>
<b>QUAECOPY</b> copies a quadrature rule into user arrays Z and W.
</li>
<li>
<b>QUAEINSIDE</b> checks whether a point is inside a triangle.
</li>
<li>
<b>QUAENODES2</b> expands nodes from 1/6 to 1/3 of the triangle.
</li>
<li>
<b>QUAENODES</b> expands nodes to the reference triangle.
</li>
<li>
<b>QUAEQUAD0</b> returns the requested quadrature rule.
</li>
<li>
<b>QUAEQUAD</b> returns a symmetric quadrature formula for a reference triangle.
</li>
<li>
<b>QUAEROTATE</b> applies a rotation.
</li>
<li>
<b>R8VEC_COPY</b> copies an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>REF_TO_KOORN</b> maps points from the reference to Koornwinder's triangle.
</li>
<li>
<b>REF_TO_TRIANGLE</b> maps points from the reference triangle to a triangle.
</li>
<li>
<b>RULE_COMPRESSED_SIZE</b> returns the compressed size of the requested quadrature rule.
</li>
<li>
<b>RULE_FULL_SIZE</b> returns the full size of the requested quadrature rule.
</li>
<li>
<b>RULE01</b> returns the rule of degree 1.
</li>
<li>
<b>RULE02</b> returns the rule of degree 2.
</li>
<li>
<b>RULE03</b> returns the rule of degree 3.
</li>
<li>
<b>RULE04</b> returns the rule of degree 4.
</li>
<li>
<b>RULE05</b> returns the rule of degree 5.
</li>
<li>
<b>RULE06</b> returns the rule of degree 6.
</li>
<li>
<b>RULE07</b> returns the rule of degree 7.
</li>
<li>
<b>RULE08</b> returns the rule of degree 8.
</li>
<li>
<b>RULE09</b> returns the rule of degree 9.
</li>
<li>
<b>RULE10</b> returns the rule of degree 10.
</li>
<li>
<b>RULE11</b> returns the rule of degree 11.
</li>
<li>
<b>RULE12</b> returns the rule of degree 12.
</li>
<li>
<b>RULE13</b> returns the rule of degree 13.
</li>
<li>
<b>RULE14</b> returns the rule of degree 14.
</li>
<li>
<b>RULE15</b> returns the rule of degree 15.
</li>
<li>
<b>RULE16</b> returns the rule of degree 16.
</li>
<li>
<b>RULE17</b> returns the rule of degree 17.
</li>
<li>
<b>RULE18</b> returns the rule of degree 18.
</li>
<li>
<b>RULE19</b> returns the rule of degree 19.
</li>
<li>
<b>RULE20</b> returns the rule of degree 20.
</li>
<li>
<b>RULE21</b> returns the rule of degree 21.
</li>
<li>
<b>RULE22</b> returns the rule of degree 22.
</li>
<li>
<b>RULE23</b> returns the rule of degree 23.
</li>
<li>
<b>RULE24</b> returns the rule of degree 24.
</li>
<li>
<b>RULE25</b> returns the rule of degree 25.
</li>
<li>
<b>RULE26</b> returns the rule of degree 26.
</li>
<li>
<b>RULE27</b> returns the rule of degree 29.
</li>
<li>
<b>RULE28</b> returns the rule of degree 28.
</li>
<li>
<b>RULE29</b> returns the rule of degree 29.
</li>
<li>
<b>RULE30</b> returns the rule of degree 30.
</li>
<li>
<b>RULE31</b> returns the rule of degree 31.
</li>
<li>
<b>RULE32</b> returns the rule of degree 32.
</li>
<li>
<b>RULE33</b> returns the rule of degree 33.
</li>
<li>
<b>RULE34</b> returns the rule of degree 34.
</li>
<li>
<b>RULE35</b> returns the rule of degree 35.
</li>
<li>
<b>RULE36</b> returns the rule of degree 36.
</li>
<li>
<b>RULE37</b> returns the rule of degree 37.
</li>
<li>
<b>RULE38</b> returns the rule of degree 38.
</li>
<li>
<b>RULE39</b> returns the rule of degree 39.
</li>
<li>
<b>RULE40</b> returns the rule of degree 40.
</li>
<li>
<b>RULE41</b> returns the rule of degree 41.
</li>
<li>
<b>RULE42</b> returns the rule of degree 42.
</li>
<li>
<b>RULE43</b> returns the rule of degree 43.
</li>
<li>
<b>RULE44</b> returns the rule of degree 44.
</li>
<li>
<b>RULE45</b> returns the rule of degree 45.
</li>
<li>
<b>RULE46</b> returns the rule of degree 46.
</li>
<li>
<b>RULE47</b> returns the rule of degree 47.
</li>
<li>
<b>RULE48</b> returns the rule of degree 48.
</li>
<li>
<b>RULE49</b> returns the rule of degree 49.
</li>
<li>
<b>RULE50</b> returns the rule of degree 50.
</li>
<li>
<b>SIMPLEX_TO_TRIANGLE</b> maps points from the simplex to a triangle.
</li>
<li>
<b>TIMESTAMP</b> prints out the current YMDHMS date as a timestamp.
</li>
<li>
<b>TRIANGLE_AREA</b> returns the area of a triangle.
</li>
<li>
<b>TRIANGLE_TO_REF</b> maps points from any triangle to the reference triangle.
</li>
<li>
<b>TRIANGLE_TO_SIMPLEX</b> maps points from any triangle to the simplex.
</li>
<li>
<b>TRIANMAP</b> maps rules from the reference triangle to the user triangle.
</li>
<li>
<b>TRIASIMP</b> maps a point from the reference triangle to the simplex.
</li>
<li>
<b>TRIASYMQ</b> returns a symmetric quadrature formula for a user triangle.
</li>
<li>
<b>TRIASYMQ_GNUPLOT:</b> set up a GNUPLOT plot of the triangle quadrature rule.
</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 30 June 2014.
</i>
<!-- John Burkardt -->
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