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<html>
<head>
<title>
TRIANGLE_LYNESS_RULE - Quadrature Rules for the Triangle
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGLE_LYNESS_RULE <br> Quadrature Rules for the Triangle
</h1>
<hr>
<p>
<b>TRIANGLE_LYNESS_RULE</b>
is a C++ library which
produces the Lyness-Jespersen family of quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
The rules have the following orders (number of points) and
precisions (maximum degree of polynomials whose integrals they
can compute exactly):
<table border="1" align="center">
<tr>
<th>Rule</th><th>Order</th><th>Precision</th>
</tr>
<tr>
<td>0</td><td> 1</td><td> 1</td>
</tr>
<tr>
<td>1</td><td> 3</td><td> 2</td>
</tr>
<tr>
<td>2</td><td> 4</td><td> 2</td>
</tr>
<tr>
<td>3</td><td> 4</td><td> 3</td>
</tr>
<tr>
<td>4</td><td> 7</td><td> 3</td>
</tr>
<tr>
<td>5</td><td> 6</td><td> 4</td>
</tr>
<tr>
<td>6</td><td> 10</td><td> 4</td>
</tr>
<tr>
<td>7</td><td> 9</td><td> 4</td>
</tr>
<tr>
<td>8</td><td> 7</td><td> 5</td>
</tr>
<tr>
<td>9</td><td> 10</td><td> 5</td>
</tr>
<tr>
<td>10</td><td> 12</td><td> 6</td>
</tr>
<tr>
<td>11</td><td> 16</td><td> 6</td>
</tr>
<tr>
<td>12</td><td> 13</td><td> 6</td>
</tr>
<tr>
<td>13</td><td> 13</td><td> 7</td>
</tr>
<tr>
<td>14</td><td> 16</td><td> 7</td>
</tr>
<tr>
<td>15</td><td> 16</td><td> 8</td>
</tr>
<tr>
<td>16</td><td> 21</td><td> 8</td>
</tr>
<tr>
<td>17</td><td> 16</td><td> 8</td>
</tr>
<tr>
<td>18</td><td> 19</td><td> 9</td>
</tr>
<tr>
<td>19</td><td> 22</td><td> 9</td>
</tr>
<tr>
<td>20</td><td> 27</td><td>11</td>
</tr>
<tr>
<td>21</td><td> 28</td><td>11</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>TRIANGLE_LYNESS_RULE</b> is available in
<a href = "../../cpp_src/triangle_lyness_rule/triangle_lyness_rule.html">a C++ version</a> and
<a href = "../../f_src/triangle_lyness_rule/triangle_lyness_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangle_lyness_rule/triangle_lyness_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 M-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_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/triangle_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a C++ 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 = "../../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>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "triangle_lyness_rule.cpp">triangle_lyness_rule.cpp</a>, the source code.
</li>
<li>
<a href = "triangle_lyness_rule.hpp">triangle_lyness_rule.hpp</a>, the include file.
</li>
<li>
<a href = "triangle_lyness_rule.sh">triangle_lyness_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_lyness_rule_prb.cpp">triangle_lyness_rule_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "triangle_lyness_rule_prb.sh">triangle_lyness_rule_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "triangle_lyness_rule_prb_output.txt">triangle_lyness_rule_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>I4_MAX</b> returns the maximum of two I4's.
</li>
<li>
<b>I4_MIN</b> returns the minimum 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>I4VEC_COPY_NEW</b> copies an I4VEC to a "new" I4VEC.
</li>
<li>
<b>LYNESS_ORDER</b> returns the order of a Lyness quadrature rule.
</li>
<li>
<b>LYNESS_PRECISION</b> returns the precision of a Lyness quadrature rule.
</li>
<li>
<b>LYNESS_RULE</b> returns the points and weights of a Lyness quadrature rule.
</li>
<li>
<b>LYNESS_RULE_NUM</b> returns the number of Lyness quadrature rules.
</li>
<li>
<b>LYNESS_SUBORDER</b> returns the suborders for a Lyness rule.
</li>
<li>
<b>LYNESS_SUBORDER_NUM</b> returns the number of suborders for a Lyness rule.
</li>
<li>
<b>LYNESS_SUBRULE</b> returns a compressed Lyness rule.
</li>
<li>
<b>R8_ABS</b> returns the absolute value of an R8.
</li>
<li>
<b>R8MAT_COPY</b> copies one R8MAT to another.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_COPY</b> copies an R8VEC.
</li>
<li>
<b>R8VEC_SUM</b> returns the sum of an R8VEC.
</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 30 September 2010.
</i>
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