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lebesgue.html
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<html>
<head>
<title>
LEBESGUE - Estimate the Lebesgue Constant
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LEBESGUE <br> Estimate the Lebesgue Constant
</h1>
<hr>
<p>
<b>LEBESGUE</b>
is a C++ 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, and which uses gnuplot to
make plots of the Lebesgue function.
</p>
<p>
Any set of nodes in the real line X(I), for 1 <= I <= N, defines
a corresponding set of Lagrange basis functions:
<pre>
L(I)(X) = product ( 1 <= J <= N, J /= I ) ( X - X(J) )
/ product ( 1 <= J <= N, J /= I ) ( X(I) - X(J) )
</pre>
with the property that
<pre>
L(I)(X(J)) = 0 if I /= J
1 if I = J
</pre>
</p>
<p>
The Lebesgue function is formed by the sum of the absolute values of
these Lagrange basis functions:
<pre>
LF(X) = sum ( 1 <= I <= N ) | L(I)(X) |
</pre>
and the Lebesgue constant LC is the maximum value of LF(X) over the
interpolation interval, which is typically X(1) to X(N), or
min ( X(*) ), max ( X(*) ), or [-1,+1], or some user-defined interval.
</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>LEBESGUE</b> is available in
<a href = "../../c_src/lebesgue/lebesgue.html">a C version</a> and
<a href = "../../cpp_src/lebesgue/lebesgue.html">a C++ version</a> and
<a href = "../../f77_src/lebesgue/lebesgue.html">a FORTRAN77 version</a> and
<a href = "../../f_src/lebesgue/lebesgue.html">a FORTRAN90 version</a> and
<a href = "../../m_src/lebesgue/lebesgue.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/gnuplot/gnuplot.html">
GNUPLOT</a>,
C++ programs which
illustrate how a program can write data and command files
so that gnuplot can create plots of the program results.
</p>
<p>
<a href = "../../cpp_src/interp/interp.html">
INTERP</a>,
a C++ library which
can be used for parameterizing and interpolating data;
</p>
<p>
<a href = "../../cpp_src/quadrule/quadrule.html">
QUADRULE</a>,
a C++ library which
defines quadrature rules for approximating an integral over a 1D domain.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Jean-Paul Berrut, Lloyd Trefethen,<br>
Barycentric Lagrange Interpolation,<br>
SIAM Review,<br>
Volume 46, Number 3, September 2004, pages 501-517.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lebesgue.cpp">lebesgue.cpp</a>, the source code.
</li>
<li>
<a href = "lebesgue.hpp">lebesgue.hpp</a>, the source code.
</li>
<li>
<a href = "lebesgue.sh">lebesgue.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "lebesgue_prb.cpp">lebesgue_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "lebesgue_prb.sh">lebesgue_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "lebesgue_prb_output.txt">lebesgue_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
The test programs create plots of the Lebesgue function for
specific sizes of the point sets.
<ul>
<li>
<a href = "chebyshev1.png">chebyshev1.png</a>
</li>
<li>
<a href = "chebyshev2.png">chebyshev2.png</a>
</li>
<li>
<a href = "chebyshev3.png">chebyshev3.png</a>
</li>
<li>
<a href = "chebyshev4.png">chebyshev4.png</a>
</li>
<li>
<a href = "equidistant1.png">equidistant1.png</a>
</li>
<li>
<a href = "equidistant2.png">equidistant2.png</a>
</li>
<li>
<a href = "equidistant3.png">equidistant3.png</a>
</li>
<li>
<a href = "fejer1.png">fejer1.png</a>
</li>
<li>
<a href = "fejer2.png">fejer2.png</a>
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>CHEBYSHEV1</b> returns the Type 1 Chebyshev points.
</li>
<li>
<b>CHEBYSHEV2</b> returns the Type 2 Chebyshev points.
</li>
<li>
<b>CHEBYSHEV3</b> returns the Type 3 Chebyshev points.
</li>
<li>
<b>CHEBYSHEV4</b> returns the Type 4 Chebyshev points.
</li>
<li>
<b>EQUIDISTANT1</b> returns the Type 1 Equidistant points.
</li>
<li>
<b>EQUIDISTANT2</b> returns the Type 2 Equidistant points.
</li>
<li>
<b>EQUIDISTANT3</b> returns the Type 3 Equidistant points.
</li>
<li>
<b>FEJER1</b> returns the Type 1 Fejer points.
</li>
<li>
<b>FEJER2</b> returns the Type 2 Fejer points.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>LAGRANGE_VALUE</b> evaluates the Lagrange polynomials.
</li>
<li>
<b>LEBESGUE_CONSTANT</b> estimates the Lebesgue constant for a set of points.
</li>
<li>
<b>LEBESGUE_FUNCTION</b> evaluates the Lebesgue function for a set of points.
</li>
<li>
<b>LEBESGUE_PLOT</b> plots the Lebesgue function for a set of points.
</li>
<li>
<b>R8VEC_PRINT</b> prints 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 = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 04 March 2014.
</i>
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