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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 MATLAB 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 = "../../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/gnuplot/gnuplot.html">
GNUPLOT</a>,
MATLAB 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 = "../../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/square_symq_rule/square_symq_rule.html">
SQUARE_SYMQ_RULE</a>,
a MATLAB 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 = "../../m_src/stroud/stroud.html">
STROUD</a>,
a MATLAB 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 = "../../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/toms886/toms886.html">
TOMS886</a>,
a MATLAB 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 = "../../m_src/triangle_dunavant_rule/triangle_dunavant_rule.html">
TRIANGLE_DUNAVANT_RULE</a>,
a MATLAB library which
sets up a Dunavant 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_integrals/triangle_integrals.html">
TRIANGLE_INTEGRALS</a>,
a MATLAB 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 = "../../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 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 = "../../m_src/triangle_ncc_rule/triangle_ncc_rule.html">
TRIANGLE_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes closed quadrature
rules on a triangle.
</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 quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a MATLAB library which
sets up a quadrature rule of exactness 5, 10, 15, 20, 25 or 30
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>
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 = "kjacopols2.m">kjacopols2.m</a>,
evaluates Jacobi polynomials and derivatives.
</li>
<li>
<a href = "kjacopols.m">kjacopols.m</a>,
evaluates Jacobi polynomials.
</li>
<li>
<a href = "klegeypols3.m">klegeypols3.m</a>,
evaluate scaled Legendre polynomials and derivatives.
</li>
<li>
<a href = "klegeypols.m">klegeypols.m</a>,
evaluates scaled Legendre polynomials.
</li>
<li>
<a href = "ortho2eva0.m">ortho2eva0.m</a>,
evaluates the orthonormal polynomials on the triangle.
</li>
<li>
<a href = "ortho2eva30.m">ortho2eva30.m</a>,
orthonormal polynomials and derivatives on triangle.
</li>
<li>
<a href = "ortho2eva3.m">ortho2eva3.m</a>,
orthogonal polynomial values and derivatives, reference triangle.
</li>
<li>
<a href = "ortho2eva.m">ortho2eva.m</a>,
evaluates orthogonal polynomials on the reference triangle.
</li>
<li>
<a href = "quaecopy2.m">quaecopy2.m</a>,
copies a quadrature rule into a user arrays X, Y, and W.
</li>
<li>
<a href = "quaecopy.m">quaecopy.m</a>,
copies a quadrature rule into user arrays Z and W.
</li>
<li>
<a href = "quaeinside.m">quaeinside.m</a>,
checks whether a point is inside a triangle.
</li>
<li>
<a href = "quaenodes2.m">quaenodes2.m</a>,
expands nodes from 1/6 to 1/3 of the triangle.
</li>
<li>
<a href = "quaenodes.m">quaenodes.m</a>,
expands nodes to the reference triangle.
</li>
<li>
<a href = "quaequad0.m">quaequad0.m</a>,
returns the requested quadrature rule.
</li>
<li>
<a href = "quaequad.m">quaequad.m</a>,
returns a symmetric quadrature formula for a reference triangle.
</li>
<li>
<a href = "quaerotate.m">quaerotate.m</a>,
applies a rotation.
</li>
<li>
<a href = "r8vec_uniform_01.m">r8vec_uniform_01.m</a>,
returns a unit pseudorandom R8VEC.
</li>
<li>
<a href = "ref_to_koorn.m">ref_to_koorn.m</a>,
maps points from the reference to Koornwinder's triangle.
</li>
<li>
<a href = "ref_to_triangle.m">ref_to_triangle.m</a>,
maps points from the reference triangle to a triangle.
</li>
<li>
<a href = "rule_compressed_size.m">rule_compressed_size.m</a>,
returns the compressed size of the requested quadrature rule.
</li>
<li>
<a href = "rule_full_size.m">rule_full_size.m</a>,
returns the full size of the requested quadrature rule.
</li>
<li>
<a href = "rule01.m">rule01.m</a>,
returns the rule of degree 1.
</li>
<li>
<a href = "rule02.m">rule02.m</a>,
returns the rule of degree 2.
</li>
<li>
<a href = "rule03.m">rule03.m</a>,
returns the rule of degree 3.
</li>
<li>
<a href = "rule04.m">rule04.m</a>,
returns the rule of degree 4.
</li>
<li>
<a href = "rule05.m">rule05.m</a>,
returns the rule of degree 5.
</li>
<li>
<a href = "rule06.m">rule06.m</a>,
returns the rule of degree 6.
</li>
<li>
<a href = "rule07.m">rule07.m</a>,
returns the rule of degree 7.
</li>
<li>
<a href = "rule08.m">rule08.m</a>,
returns the rule of degree 8.
</li>
<li>
<a href = "rule09.m">rule09.m</a>,
returns the rule of degree 9.
</li>
<li>
<a href = "rule10.m">rule10.m</a>,
returns the rule of degree 10.
</li>
<li>
<a href = "rule11.m">rule11.m</a>,
returns the rule of degree 11.
</li>
<li>
<a href = "rule12.m">rule12.m</a>,
returns the rule of degree 12.
</li>
<li>
<a href = "rule13.m">rule13.m</a>,
returns the rule of degree 13.
</li>
<li>
<a href = "rule14.m">rule14.m</a>,
returns the rule of degree 14.
</li>
<li>
<a href = "rule15.m">rule15.m</a>,
returns the rule of degree 15.
</li>
<li>
<a href = "rule16.m">rule16.m</a>,
returns the rule of degree 16.
</li>
<li>
<a href = "rule17.m">rule17.m</a>,
returns the rule of degree 17.
</li>
<li>
<a href = "rule18.m">rule18.m</a>,
returns the rule of degree 18.
</li>
<li>
<a href = "rule19.m">rule19.m</a>,
returns the rule of degree 19.
</li>
<li>
<a href = "rule20.m">rule20.m</a>,
returns the rule of degree 20.
</li>
<li>
<a href = "rule21.m">rule21.m</a>,
returns the rule of degree 21.
</li>
<li>
<a href = "rule22.m">rule22.m</a>,
returns the rule of degree 22.
</li>
<li>
<a href = "rule23.m">rule23.m</a>,
returns the rule of degree 23.
</li>
<li>
<a href = "rule24.m">rule24.m</a>,
returns the rule of degree 24.
</li>
<li>
<a href = "rule25.m">rule25.m</a>,
returns the rule of degree 25.
</li>
<li>
<a href = "rule26.m">rule26.m</a>,
returns the rule of degree 26.
</li>
<li>
<a href = "rule27.m">rule27.m</a>,
returns the rule of degree 29.
</li>
<li>
<a href = "rule28.m">rule28.m</a>,
returns the rule of degree 28.
</li>
<li>
<a href = "rule29.m">rule29.m</a>,
returns the rule of degree 29.
</li>
<li>
<a href = "rule30.m">rule30.m</a>,
returns the rule of degree 30.
</li>
<li>
<a href = "rule31.m">rule31.m</a>,
returns the rule of degree 31.
</li>
<li>
<a href = "rule32.m">rule32.m</a>,
returns the rule of degree 32.
</li>
<li>
<a href = "rule33.m">rule33.m</a>,
returns the rule of degree 33.
</li>
<li>
<a href = "rule34.m">rule34.m</a>,
returns the rule of degree 34.
</li>
<li>
<a href = "rule35.m">rule35.m</a>,
returns the rule of degree 35.
</li>
<li>
<a href = "rule36.m">rule36.m</a>,
returns the rule of degree 36.
</li>
<li>
<a href = "rule37.m">rule37.m</a>,
returns the rule of degree 37.
</li>
<li>
<a href = "rule38.m">rule38.m</a>,
returns the rule of degree 38.
</li>
<li>
<a href = "rule39.m">rule39.m</a>,
returns the rule of degree 39.
</li>
<li>
<a href = "rule40.m">rule40.m</a>,
returns the rule of degree 40.
</li>
<li>
<a href = "rule41.m">rule41.m</a>,
returns the rule of degree 41.
</li>
<li>
<a href = "rule42.m">rule42.m</a>,
returns the rule of degree 42.
</li>
<li>
<a href = "rule43.m">rule43.m</a>,
returns the rule of degree 43.
</li>
<li>
<a href = "rule44.m">rule44.m</a>,
returns the rule of degree 44.
</li>
<li>
<a href = "rule45.m">rule45.m</a>,
returns the rule of degree 45.
</li>
<li>
<a href = "rule46.m">rule46.m</a>,
returns the rule of degree 46.
</li>
<li>
<a href = "rule47.m">rule47.m</a>,
returns the rule of degree 47.
</li>
<li>
<a href = "rule48.m">rule48.m</a>,
returns the rule of degree 48.
</li>
<li>
<a href = "rule49.m">rule49.m</a>,
returns the rule of degree 49.
</li>
<li>
<a href = "rule50.m">rule50.m</a>,
the rule of degree 50.
</li>
<li>
<a href = "simplex_to_triangle.m">simplex_to_triangle.m</a>,
maps points from the simplex to a triangle.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints out the current YMDHMS date as a timestamp.
</li>
<li>
<a href = "triangle_area.m">triangle_area.m</a>,
returns the area of a triangle.
</li>
<li>
<a href = "triangle_to_ref.m">triangle_to_ref.m</a>,
maps points from any triangle to the reference triangle.
</li>
<li>
<a href = "triangle_to_simplex.m">triangle_to_simplex.m</a>,
maps points from any triangle to the simplex.
</li>
<li>
<a href = "trianmap.m">trianmap.m</a>,
maps rules from the reference triangle to the user triangle.
</li>
<li>
<a href = "triasimp.m">triasimp.m</a>,
maps a point from the reference triangle to the simplex.
</li>
<li>
<a href = "triasymq.m">triasymq.m</a>,
returns a symmetric quadrature formula for a user triangle.
</li>
<li>
<a href = "triasymq_gnuplot.m">triasymq_gnuplot.m</a>,
set up a GNUPLOT plot of the triangle quadrature rule.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "triangle_symq_rule_test.m">triangle_symq_rule_test.m</a>,
a sample calling program.
</li>
<li>
<a href = "triangle_symq_rule_test01.m">triangle_symq_rule_test01.m</a>,
tests TRIANGLE_TO_SIMPLEX, TRIANGLE_TO_REF, REF_TO_TRIANGLE, SIMPLEX_TO_TRIANGLE.
</li>
<li>
<a href = "triangle_symq_rule_test02.m">triangle_symq_rule_test02.m</a>,
calls TRIASYMQ for a quadrature rule of given order and region.
</li>
<li>
<a href = "triangle_symq_rule_test03.m">triangle_symq_rule_test03.m</a>,
calls TRIASYMQ_GNUPLOT to generate graphics files.
</li>
<li>
<a href = "triangle_symq_rule_test04.m">triangle_symq_rule_test04.m</a>,
gets a rule and writes it to a file.
</li>
<li>
<a href = "triangle_symq_rule_test05.m">triangle_symq_rule_test05.m</a>,
calls TRIASYMQ for a quadrature rule of given order and region.
</li>
<li>
<a href = "triangle_symq_rule_test_output.txt">
triangle_symq_rule_test_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>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 27 June 2014.
</i>
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