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<html>
<head>
<title>
TETRAHEDRON_KEAST_RULE - Quadrature Rules for a Tetrahedron
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TETRAHEDRON_KEAST_RULE <br> Quadrature Rules for a Tetrahedron
</h1>
<hr>
<p>
<b>TETRAHEDRON_KEAST_RULE</b>
is a C++ library which
defines ten quadrature rules, of degree of exactness 0 through 8,
over the interior of the tetrahedron in 3D.
</p>
<p>
The ten rules have the following orders and precisions:
<table border="1" align="center">
<tr>
<th>Rule</th><th>Order</th><th>Precision</th>
</tr>
<tr>
<td> 1</td><td> 1</td><td> 0</td>
</tr>
<tr>
<td> 2</td><td> 4</td><td> 1</td>
</tr>
<tr>
<td> 3</td><td> 5</td><td> 2</td>
</tr>
<tr>
<td> 4</td><td>10</td><td> 3</td>
</tr>
<tr>
<td> 5</td><td>11</td><td> 4</td>
</tr>
<tr>
<td> 6</td><td>14</td><td> 4</td>
</tr>
<tr>
<td> 7</td><td>15</td><td> 5</td>
</tr>
<tr>
<td> 8</td><td>24</td><td> 6</td>
</tr>
<tr>
<td> 9</td><td>31</td><td> 7</td>
</tr>
<tr>
<td>10</td><td>45</td><td> 8</td>
</tr>
</table>
</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>TETRAHEDRON_KEAST_RULE</b> is available in
<a href = "../../cpp_src/tetrahedron_keast_rule/tetrahedron_keast_rule.html">a C++ version</a> and
<a href = "../../f_src/tetrahedron_keast_rule/tetrahedron_keast_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/tetrahedron_keast_rule/tetrahedron_keast_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 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/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_arbq_rule/tetrahedron_arbq_rule.html">
TETRAHEDRON_ARBQ_RULE</a>,
a C++ library which
returns quadrature rules,
with exactness up to total degree 15,
over the interior of a tetrahedron in 3D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_exactness/tetrahedron_exactness.html">
TETRAHEDRON_EXACTNESS</a>,
a C++ program which
investigates the polynomial exactness of a quadrature rule
over the interior of a tetrahedron in 3D.
</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/tetrahedron_integrals/tetrahedron_integrals.html">
TETRAHEDRON_INTEGRALS</a>,
a C++ library which
returns the exact value of the integral of any monomial
over the interior of the unit tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_monte_carlo/tetrahedron_monte_carlo.html">
TETRAHEDRON_MONTE_CARLO</a>,
a C++ library which
uses the Monte Carlo method to estimate integrals
over the interior of the unit tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">
TETRAHEDRON_NCC_RULE</a>,
a C++ library which
defines Newton-Cotes Closed (NCC) quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../cpp_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">
TETRAHEDRON_NCO_RULE</a>,
a C++ library which
defines Newton-Cotes Open (NCO) quadrature rules
over the interior of a tetrahedron in 3D.
</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/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>
Patrick Keast,<br>
Moderate Degree Tetrahedral Quadrature Formulas,<br>
Computer Methods in Applied Mechanics and Engineering,<br>
Volume 55, Number 3, May 1986, pages 339-348.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "tetrahedron_keast_rule.cpp">tetrahedron_keast_rule.cpp</a>, the source code.
</li>
<li>
<a href = "tetrahedron_keast_rule.hpp">tetrahedron_keast_rule.hpp</a>, the include file.
</li>
<li>
<a href = "tetrahedron_keast_rule.sh">tetrahedron_keast_rule.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "tetrahedron_keast_rule_prb.cpp">tetrahedron_keast_rule_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "tetrahedron_keast_rule_prb.sh">tetrahedron_keast_rule_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "tetrahedron_keast_rule_prb_output.txt">tetrahedron_keast_rule_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>COMP_NEXT</b> computes the compositions of the integer N into K parts.
</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>KEAST_DEGREE</b> returns the degree of a Keast rule for the triangle.
</li>
<li>
<b>KEAST_ORDER_NUM</b> returns the order of a Keast rule for the triangle.
</li>
<li>
<b>KEAST_RULE</b> returns the points and weights of a Keast rule.
</li>
<li>
<b>KEAST_RULE_NUM</b> returns the number of Keast rules available.
</li>
<li>
<b>KEAST_SUBORDER</b> returns the suborders for a Keast rule.
</li>
<li>
<b>KEAST_SUBORDER_NUM</b> returns the number of suborders for a Keast rule.
</li>
<li>
<b>KEAST_SUBRULE</b> returns a compressed Keast rule.
</li>
<li>
<b>KEAST_SUBRULE_01</b> returns a compressed Keast rule 1.
</li>
<li>
<b>KEAST_SUBRULE_02</b> returns a compressed Keast rule 2.
</li>
<li>
<b>KEAST_SUBRULE_03</b> returns a compressed Keast rule 3.
</li>
<li>
<b>KEAST_SUBRULE_04</b> returns a compressed Keast rule 4.
</li>
<li>
<b>KEAST_SUBRULE_05</b> returns a compressed Keast rule 5.
</li>
<li>
<b>KEAST_SUBRULE_06</b> returns a compressed Keast rule 6.
</li>
<li>
<b>KEAST_SUBRULE_07</b> returns a compressed Keast rule 7.
</li>
<li>
<b>KEAST_SUBRULE_08</b> returns a compressed Keast rule 8.
</li>
<li>
<b>KEAST_SUBRULE_08</b> returns a compressed Keast rule 8.
</li>
<li>
<b>KEAST_SUBRULE_10</b> returns a compressed Keast rule 10.
</li>
<li>
<b>MONOMIAL_VALUE</b> evaluates a monomial.
</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>R8MAT_DET_4D</b> computes the determinant of a 4 by 4 R8MAT.
</li>
<li>
<b>R8VEC_DOT</b> computes the dot product of a pair of R8VEC's.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>TETRAHEDRON_REFERENCE_TO_PHYSICAL</b> maps reference points to physical points.
</li>
<li>
<b>TETRAHEDRON_VOLUME</b> computes the volume of a tetrahedron in 3D.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</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 27 June 2014.
</i>
<!-- John Burkardt -->
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