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<html>
<head>
<title>
LINE_FEKETE_RULE - Approximate Fekete Points in an Interval
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LINE_FEKETE_RULE<br>
Approximate Fekete Points in an Interval
</h1>
<hr>
<p>
<b>LINE_FEKETE_RULE</b>
is 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>
Given a desired number of points N, the best choice for abscissas is
a set of Lebesgue points, which minimize the Lebesgue constant,
which describes the error in polynomial interpolation. Sets of
Lebesgue points are difficult to define mathematically. Fekete
points are a related, computable set, defined as those sets maximizing
the magnitude of the determinant of the Vandermonde matrix associated
with the points. Analytic definitions of these points are known for
a few cases, but there is a general computational procedure for
approximating them, which is demonstrated here.
</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>LINE_FEKETE_RULE</b> is available in
<a href = "../../c_src/line_fekete_rule/line_fekete_rule.html">a C version</a> and
<a href = "../../cpp_src/line_fekete_rule/line_fekete_rule.html">a C++ version</a> and
<a href = "../../f77_src/line_fekete_rule/line_fekete_rule.html">a FORTRAN77 version</a> and
<a href = "../../f_src/line_fekete_rule/line_fekete_rule.html">a FORTRAN90 version</a> and
<a href = "../../m_src/line_fekete_rule/line_fekete_rule.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/lebesgue/lebesgue.html">
LEBESGUE</a>,
a MATLAB library which
is given a set of nodes in 1D, and
plots the Lebesgue function, and estimates the Lebesgue constant,
which measures the maximum magnitude of the potential error
of Lagrange polynomial interpolation.
</p>
<p>
<a href = "../../m_src/line_felippa_rule/line_felippa_rule.html">
LINE_FELIPPA_RULE</a>,
a MATLAB library which
returns the points and weights of a Felippa quadrature rule
over the interior of a line segment in 1D.
</p>
<p>
<a href = "../../m_src/line_grid/line_grid.html">
LINE_GRID</a>,
a MATLAB library which
computes a grid of points
over the interior of a line segment in 1D.
</p>
<p>
<a href = "../../m_src/line_ncc_rule/line_ncc_rule.html">
LINE_NCC_RULE</a>,
a MATLAB library which
computes a Newton Cotes Closed (NCC) quadrature rule for the line,
that is, for an interval of the form [A,B], using equally spaced points
which include the endpoints.
</p>
<p>
<a href = "../../m_src/line_nco_rule/line_nco_rule.html">
LINE_NCO_RULE</a>,
a MATLAB library which
computes a Newton Cotes Open (NCO) quadrature rule,
using equally spaced points,
over the interior of a line segment in 1D.
</p>
<p>
<a href = "../../m_src/quadrature_weights_vandermonde/quadrature_weights_vandermonde.html">
QUADRATURE_WEIGHTS_VANDERMONDE</a>,
a MATLAB library which
computes the weights of a quadrature rule using the Vandermonde
matrix, assuming that the points have been specified.
</p>
<p>
<a href = "../../m_src/triangle_fekete_rule/triangle_fekete_rule.html">
TRIANGLE_FEKETE_RULE</a>,
a MATLAB library which
defines Fekete rules for quadrature or interpolation over a triangle.
</p>
<p>
<a href = "../../m_src/vandermonde/vandermonde.html">
VANDERMONDE</a>,
a MATLAB library which
carries out certain operations associated
with the Vandermonde matrix.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Len Bos, Norm Levenberg,<br>
On the calculation of approximate Fekete points:
the univariate case,<br>
Electronic Transactions on Numerical Analysis, <br>
Volume 30, pages 377-397, 2008.
</li>
<li>
Alvise Sommariva, Marco Vianello,<br>
Computing approximate Fekete points by QR factorizations of Vandermonde
matrices,<br>
Computers and Mathematics with Applications,<br>
Volume 57, 2009, pages 1324-1336.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "chebvand.m">chebvand.m</a>
returns the Chebyshev Vandermonde matrix,
as programmed by Nick Higham.
</li>
<li>
<a href = "cheby_van1.m">cheby_van1.m</a>
returns the Chebyshev Vandermonde matrix.
</li>
<li>
<a href = "legendre_van.m">legendre_van.m</a>
returns the Legendre Vandermonde matrix.
</li>
<li>
<a href = "line_fekete_chebyshev.m">line_fekete_chebyshev.m</a>
approximates Fekete points, using the Chebyshev polynomials
and the Chebyshev weight function.
</li>
<li>
<a href = "line_fekete_monomial.m">line_fekete_monomial.m</a>
approximates Fekete points, using the monomials
and the uniform weight function.
</li>
<li>
<a href = "line_monomial_moments.m">line_monomial_moments.m</a>
returns the moments of monomials over [A,B].
</li>
<li>
<a href = "r8vec_print.m">r8vec_print.m</a>
prints an R8VEC.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints the YMDHMS date as a timestamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "line_fekete_rule_test.m">line_fekete_rule_test.m</a>
runs all the tests.
</li>
<li>
<a href = "line_fekete_rule_test_output.txt">line_fekete_rule_test_output.txt</a>
the output file.
</li>
<li>
<a href = "line_fekete_rule_test01.m">line_fekete_rule_test01.m</a>
tests line_fekete_monomial().
</li>
<li>
<a href = "line_fekete_rule_test02.m">line_fekete_rule_test02.m</a>
tests line_fekete_chebyshev().
</li>
<li>
<a href = "line_fekete_rule_test02.png">line_fekete_rule_test02.png</a>
an image of the location of the Fekete points for N = 21.
</li>
<li>
<a href = "line_fekete_rule_bos_levenberg_test.m">line_fekete_rule_bos_levenberg_test.m</a>
computes the same items as line_fekete_chebyshev, but using
code directly from the Bos-Levenberg paper.
</li>
<li>
<a href = "line_fekete_rule_bos_levenberg_test.png">line_fekete_rule_bos_levenberg_test.pngm</a>
an image of the location of the Fekete points for N = 21.
</li>
<li>
<a href = "line_fekete_rule_test03.m">line_fekete_rule_test03.m</a>
tests line_fekete_legendre().
</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 24 March 2014.
</i>
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