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<html>
<head>
<title>
FEKETE - High Order Interpolation and Quadrature in Triangles
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEKETE <br> High Order Interpolation and Quadrature in Triangles
</h1>
<hr>
<p>
<b>FEKETE</b>
is a FORTRAN90 library which
can return information defining any of seven Fekete
rules for high order interpolation and quadrature in a triangle.
</p>
<p>
Fekete points can be defined for any region OMEGA. To define
the Fekete points for a given region, let Poly(N) be some finite
dimensional vector space of polynomials, such as all polynomials
of degree less than L, or all polynomials whose monomial terms
have total degree less than some value L.
</p>
<p>
Let P(1:M) be any basis for Poly(N). For this basis, the Fekete
points are defined as those points Z(1:M) which maximize the
determinant of the corresponding Vandermonde matrix:
<pre>
V = [ P1(Z1) P1(Z2) ... P1(ZM) ]
[ P2(Z1) P2(Z2) ... P2(ZM) ]
...
[ PM(ZM) P2(ZM) ... PM(ZM) ]
</pre>
</p>
<p>
On the triangle, it is known that some Fekete points will lie
on the boundary, and that on each side of the triangle, these
points will correspond to a set of Gauss-Lobatto points.
</p>
<p>
The seven 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> 10</td><td> 3</td>
</tr>
<tr>
<td>2</td><td> 28</td><td> 6</td>
</tr>
<tr>
<td>3</td><td> 55</td><td> 9</td>
</tr>
<tr>
<td>4</td><td> 91</td><td>12</td>
</tr>
<tr>
<td>5</td><td> 91</td><td>12</td>
</tr>
<tr>
<td>6</td><td>136</td><td>15</td>
</tr>
<tr>
<td>7</td><td>190</td><td>18</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>FEKETE</b> is available in
<a href = "../../cpp_src/fekete/fekete.html">a C++ version</a> and
<a href = "../../f_src/fekete/fekete.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fekete/fekete.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/dunavant/dunavant.html">
DUNAVANT</a>,
a FORTRAN90 library which
defines Dunavant rules for quadrature
on a triangle.
</p>
<p>
<a href = "../../f_src/felippa/felippa.html">
FELIPPA</a>,
a FORTRAN90 library which
defines quadrature rules for lines, triangles, quadrilaterals,
pyramids, wedges, tetrahedrons and hexahedrons.
</p>
<p>
<a href = "../../f_src/gm_rule/gm_rule.html">
GM_RULE</a>,
a FORTRAN90 library which
defines a Grundmann-Moeller
rule for quadrature over a triangle, tetrahedron, or general
M-dimensional simplex.
</p>
<p>
<a href = "../../f_src/lyness_rule/lyness_rule.html">
LYNESS_RULE</a>
a FORTRAN90 library which
returns Lyness-Jespersen quadrature rules for the triangle.
</p>
<p>
<a href = "../../f_src/ncc_triangle/ncc_triangle.html">
NCC_TRIANGLE</a>,
a FORTRAN90 library which
defines Newton-Cotes closed quadrature
rules on a triangle.
</p>
<p>
<a href = "../../f_src/nco_triangle/nco_triangle.html">
NCO_TRIANGLE</a>,
a FORTRAN90 library which
defines Newton-Cotes open quadrature
rules on a triangle.
</p>
<p>
<a href = "../../datasets/quadrature_rules_tri/quadrature_rules_tri.html">
QUADRATURE_RULES_TRI</a>,
a dataset directory which
contains triples of files which
defines various quadrature
rules on triangles.
</p>
<p>
<a href = "../../f_src/stroud/stroud.html">
STROUD</a>,
a FORTRAN90 library which
contains quadrature rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and M-dimensions.
</p>
<p>
<a href = "../../f_src/test_tri_int/test_tri_int.html">
TEST_TRI_INT</a>,
a FORTRAN90 library which
tests algorithms for quadrature over a triangle.
</p>
<p>
<a href = "../../f77_src/toms612/toms612.html">
TOMS612</a>,
a FORTRAN77 library which
estimates the integral of a function over a triangle.
</p>
<p>
<a href = "../../f_src/triangle_exactness/triangle_exactness.html">
TRIANGLE_EXACTNESS</a>,
a FORTRAN90 program which
investigates the polynomial exactness of a quadrature rule for the triangle.
</p>
<p>
<a href = "../../f_src/triangle_monte_carlo/triangle_monte_carlo.html">
TRIANGLE_MONTE_CARLO</a>,
a FORTRAN90 program which
uses the Monte Carlo method to estimate integrals over a triangle.
</p>
<p>
<a href = "../../f_src/wandzura/wandzura.html">
WANDZURA</a>,
a FORTRAN90 library which
definines Wandzura rules for quadrature on a triangle.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
SF Bockman,<br>
Generalizing the Formula for Areas of Polygons to Moments,<br>
American Mathematical Society Monthly,<br>
Volume 96, Number 2, February 1989, pages 131-132.
</li>
<li>
Hermann Engels,<br>
Numerical Quadrature and Cubature,<br>
Academic Press, 1980,<br>
ISBN: 012238850X,<br>
LC: QA299.3E5.
</li>
<li>
Arthur Stroud,<br>
Approximate Calculation of Multiple Integrals,<br>
Prentice Hall, 1971,<br>
ISBN: 0130438936,<br>
LC: QA311.S85.
</li>
<li>
Mark Taylor, Beth Wingate, Rachel Vincent,<br>
An Algorithm for Computing Fekete Points in the Triangle,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 38, Number 5, 2000, pages 1707-1720.
</li>
<li>
Stephen Wandzura, Hong Xiao,<br>
Symmetric Quadrature Rules on a Triangle,<br>
Computers and Mathematics with Applications,<br>
Volume 45, 2003, pages 1829-1840.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fekete.f90">fekete.f90</a>, the source code.
</li>
<li>
<a href = "fekete.sh">fekete.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "fekete_prb.f90">fekete_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "fekete_prb.sh">fekete_prb.sh</a>,
commands to compile and run the sample program.
</li>
<li>
<a href = "fekete_prb_output.txt">fekete_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
One of the tests in the sample calling program creates
EPS files of
the abscissas in the unit triangle. These have been converted
to PNG files for
display here.
<ul>
<li>
<a href = "fekete_rule_1.png">fekete_rule_1.png</a>,
a plot of rule 1.
</li>
<li>
<a href = "fekete_rule_2.png">fekete_rule_2.png</a>,
a plot of rule 2.
</li>
<li>
<a href = "fekete_rule_3.png">fekete_rule_3.png</a>,
a plot of rule 3.
</li>
<li>
<a href = "fekete_rule_4.png">fekete_rule_4.png</a>,
a plot of rule 4.
</li>
<li>
<a href = "fekete_rule_5.png">fekete_rule_5.png</a>,
a plot of rule 5.
</li>
<li>
<a href = "fekete_rule_6.png">fekete_rule_6.png</a>,
a plot of rule 6.
</li>
<li>
<a href = "fekete_rule_7.png">fekete_rule_7.png</a>,
a plot of rule 7.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>FEKETE_DEGREE</b> returns the degree of a given Fekete rule for the triangle.
</li>
<li>
<b>FEKETE_RULE</b> returns the points and weights of a Fekete rule.
</li>
<li>
<b>FEKETE_RULE_NUM</b> returns the number of Fekete rules available.
</li>
<li>
<b>FEKETE_ORDER_NUM</b> returns the order of a given Fekete rule for the triangle.
</li>
<li>
<b>FEKETE_SUBORDER</b> returns the suborders for a Fekete rule.
</li>
<li>
<b>FEKETE_SUBORDER_NUM</b> returns the number of suborders for a Fekete rule.
</li>
<li>
<b>FEKETE_SUBRULE</b> returns a compressed Fakete rule.
</li>
<li>
<b>FILE_NAME_INC</b> increments a partially numeric filename.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of integer division.
</li>
<li>
<b>I4_WRAP</b> forces an integer to lie between given limits by wrapping.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_T3</b> maps T3 reference points to physical points.
</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>TRIANGLE_POINTS_PLOT</b> plots a triangle and some points.
</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 28 December 2010.
</i>
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