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<html>
<head>
<title>
TRIANGLE_FEKETE_RULE - High Order Interpolation and Quadrature in Triangles
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TRIANGLE_FEKETE_RULE <br> High Order Interpolation and Quadrature in Triangles
</h1>
<hr>
<p>
<b>TRIANGLE_FEKETE_RULE</b>
is a MATLAB library which
can return information defining any of seven Fekete
rules for high order interpolation and quadrature
over the interior of a triangle in 2D.
</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>
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>
<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>
<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_FEKETE_RULE</b> is available in
<a href = "../../c_src/triangle_fekete_rule/triangle_fekete_rule.html">a C version</a> and
<a href = "../../cpp_src/triangle_fekete_rule/triangle_fekete_rule.html">a C++ version</a> and
<a href = "../../f_src/triangle_fekete_rule/triangle_fekete_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/triangle_fekete_rule/triangle_fekete_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/line_fekete/line_fekete.html">
LINE_FEKETE</a>,
a MATLAB library which
approximates the location of Fekete points in an interval [A,B].
A family of sets of Fekete points, indexed by size N, represents
an excellent choice for defining a polynomial interpolant.
</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/stroud/stroud.html">
STROUD</a>,
a MATLAB library which
contains quadrature
rules for a variety of unusual areas, surfaces and volumes in 2D,
3D and N-dimensions.
</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/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_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
implements test functions for integration
over the interior of a 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 program which
uses the Monte Carlo method to estimate integrals
over the interior of a 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
over the interior of a triangle in 2D.
</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_symq_rule/triangle_symq_rule.html">
TRIANGLE_SYMQ_RULE</a>,
a MATLAB 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 = "../../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>
<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_degree.m">fekete_degree.m</a>
returns the degree of a given Fekete rule for the triangle.
</li>
<li>
<a href = "fekete_order_num.m">fekete_order_num.m</a>
returns the order of a given Fekete rule for the triangle.
</li>
<li>
<a href = "fekete_rule.m">fekete_rule.m</a>
returns the points and weights of a Fekete rule.
</li>
<li>
<a href = "fekete_rule_num.m">fekete_rule_num.m</a>
returns the number of Fekete rules available.
</li>
<li>
<a href = "fekete_suborder.m">fekete_suborder.m</a>
returns the suborders for a Fekete rule.
</li>
<li>
<a href = "fekete_suborder_num.m">fekete_suborder_num.m</a>
returns the number of suborders for a Fekete rule.
</li>
<li>
<a href = "fekete_subrule.m">fekete_subrule.m</a>
returns a compressed Fakete rule.
</li>
<li>
<a href = "file_name_inc.m">file_name_inc.m</a>
increments a partially numeric filename.
</li>
<li>
<a href = "i4_modp.m">i4_modp.m</a>
returns the nonnegative remainder of integer division.
</li>
<li>
<a href = "i4_wrap.m">i4_wrap.m</a>
forces an integer to lie between given limits by wrapping.
</li>
<li>
<a href = "reference_to_physical_t3.m">reference_to_physical_t3.m</a>
maps T3 reference points to physical points.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints the current YMDHMS date as a time stamp.
</li>
<li>
<a href = "triangle_area.m">triangle_area.m</a>
computes the area of a triangle.
</li>
<li>
<a href = "triangle_points_plot.m">triangle_points_plot.m</a>
plots a triangle and some points.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "triangle_fekete_rule_test.m">triangle_fekete_rule_test.m</a>,
runs all the tests.
</li>
<li>
<a href = "triangle_fekete_rule_test01.m">triangle_fekete_rule_test01.m</a>,
tests FEKETE_RULE_NUM, FEKETE_DEGREE, and
FEKETE_ORDER_NUM.
</li>
<li>
<a href = "triangle_fekete_rule_test02.m">triangle_fekete_rule_test02.m</a>,
tests FEKETE_RULE by summing the weights.
</li>
<li>
<a href = "triangle_fekete_rule_test03.m">triangle_fekete_rule_test03.m</a>,
tests FEKETE_RULE by summing the barycentric coordinates.
</li>
<li>
<a href = "triangle_fekete_rule_test04.m">triangle_fekete_rule_test04.m</a>,
tests FEKETE_ORDER by integrating monomials in the
unit triangle.
</li>
<li>
<a href = "triangle_fekete_rule_test05.m">triangle_fekete_rule_test05.m</a>,
tests FEKETE_RULE by plotting the points.
</li>
<li>
<a href = "triangle_fekete_rule_test06.m">triangle_rule_fekete_test06.m</a>,
tests REFERENCE_TO_PHYSICAL_T3 by transforming a Fekete rule
from the unit triangle to another triangle.
</li>
<li>
<a href = "triangle_fekete_rule_test_output.txt">triangle_fekete_rule_test_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>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 15 March 2014.
</i>
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