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<html>
<head>
<title>
TRIANGLE_WANDZURA_RULE - High Order Quadrature in a Triangle.
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGLE_WANDZURA_RULE <br> High Order Quadrature in a Triangle.
</h1>
<hr>
<p>
<b>TRIANGLE_WANDZURA_RULE</b>
is a C++ library which
can return any of six Wandzura
rules for high order quadrature
over the interior of a triangle in 2D.
</p>
<p>
There are six rules, which have polynomial degree of exactness
of 5, 10, 15, 20, 25, and 30.
</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_WANDZURA_RULE</b> is available in
<a href = "../../cpp_src/triangle_wandzura_rule/triangle_wandzura_rule.html">a C++ version</a> and
<a href = "../../f_src/triangle_wandzura_rule/triangle_wandzura_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangle_wandzura_rule/triangle_wandzura_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
contains 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_dunavant_rule/triangle_dunavant_rule.html">
TRIANGLE_DUNAVANT_RULE</a>,
a C++ library which
sets up a Dunavant quadrature rule
over the interior of a triangle in 2D.
</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_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++ program which
uses the Monte Carlo method to estimate integrals
over the interior of a 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_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/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>
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>
<li>
Stephen Wandzura, Hong Xiao,<br>
Symmetric Quadrature Rules on a Triangle,<br>
Computers and Mathematics with Applications,<br>
Volume 45, Number 12, June 2003, pages 1829-1840.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "triangle_wandzura_rule.cpp">triangle_wandzura_rule.cpp</a>, the source code.
</li>
<li>
<a href = "triangle_wandzura_rule.hpp">triangle_wandzura_rule.hpp</a>, the include file.
</li>
<li>
<a href = "triangle_wandzura_rule.sh">triangle_wandzura_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_wandzura_rule_prb.cpp">triangle_wandzura_rule_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "triangle_wandzura_rule_prb.sh">triangle_wandzura_rule_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "triangle_wandzura_rule_prb_output.txt">triangle_wandzura_rule_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FILE_NAME_INC</b> increments a partially numeric file name.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two I4's.
</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>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>
<li>
<b>WANDZURA_DEGREE</b> returns the degree of a given Wandzura rule for the triangle.
</li>
<li>
<b>WANDZURA_ORDER_NUM</b> returns the order of a given Wandzura rule for the triangle.
</li>
<li>
<b>WANDZURA_RULE</b> returns the points and weights of a Wandzura rule.
</li>
<li>
<b>WANDZURA_RULE_NUM</b> returns the number of Wandzura rules available.
</li>
<li>
<b>WANDZURA_SUBORDER</b> returns the suborders for a Wandzura rule.
</li>
<li>
<b>WANDZURA_SUBORDER_NUM</b> returns the number of suborders for a Wandzura rule.
</li>
<li>
<b>WANDZURA_SUBRULE</b> returns a compressed Wandzura rule.
</li>
<li>
<b>WANDZURA_SUBRULE_1</b> returns a compressed Wandzura rule 1.
</li>
<li>
<b>WANDZURA_SUBRULE_2</b> returns a compressed Wandzura rule 2.
</li>
<li>
<b>WANDZURA_SUBRULE_3</b> returns a compressed Wandzura rule 3.
</li>
<li>
<b>WANDZURA_SUBRULE_4</b> returns a compressed Wandzura rule 4.
</li>
<li>
<b>WANDZURA_SUBRULE_5</b> returns a compressed Wandzura rule 5.
</li>
<li>
<b>WANDZURA_SUBRULE_6</b> returns a compressed Wandzura rule 6.
</li>
<li>
<b>WANDZURA_SUBRULE2_1</b> returns a compressed Wandzura rule 1.
</li>
<li>
<b>WANDZURA_SUBRULE2_2</b> returns a compressed Wandzura rule 2.
</li>
<li>
<b>WANDZURA_SUBRULE2_3</b> returns a compressed Wandzura rule 3.
</li>
<li>
<b>WANDZURA_SUBRULE2_4</b> returns a compressed Wandzura rule 4.
</li>
<li>
<b>WANDZURA_SUBRULE2_5</b> returns a compressed Wandzura rule 5.
</li>
<li>
<b>WANDZURA_SUBRULE2_6</b> returns a compressed Wandzura rule 6.
</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>
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