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<html>
<head>
<title>
BARYCENTRIC_INTERP_1D - Barycentric Lagrange Polynomial Interpolation in 1D
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
BARYCENTRIC_INTERP_1D <br> Barycentric Lagrange Polynomial Interpolation in 1D
</h1>
<hr>
<p>
<b>BARYCENTRIC_INTERP_1D</b>
is a C++ library which
defines and evaluates the Lagrange polynomial p(x)
which interpolates a set of data, so that p(x(i)) = y(i).
Because a barycentric formulation is used, polynomials of very high
degree can safely be used.
</p>
<p>
Efficient calculation of the barycentric polynomial interpolant requires
that the function to be interpolated be sampled at points from a known
family, for which the interpolation weights have been precomputed.
Such families include
<ul>
<li>
evenly spaced points (but this results in an ill-conditioned system);
</li>
<li>
Chebyshev Type 1 points;
</li>
<li>
Chebyshev Type 2 points;
</li>
<li>
Chebyshev Type 3 points;
</li>
<li>
Chebyshev Type 4 points;
</li>
</ul>
and any linear mapping of these points to an arbitary interval [A,B].
</p>
<p>
<i>
Note that in the Berrut/Trefethen reference, there is a significant
typographical error on page 510, where an adjustment is made in cases
where the polynomial is to be evaluated exactly at a data point.
The paper reads:
<pre>
exact(xdiff==0) = 1;
</pre>
but it should read
<pre>
exact(xdiff==0) = j;
</pre>
</i>
</p>
<p>
<b>BARYCENTRIC_INTERP_1D</b> requires the R8LIB library. The test also requires
the TEST_INTERP_1D library.
</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>BARYCENTRIC_INTERP_1D</b> is available in
<a href = "../../c_src/barycentric_interp_1d/barycentric_interp_1d.html">a C version</a> and
<a href = "../../cpp_src/barycentric_interp_1d/barycentric_interp_1d.html">a C++ version</a> and
<a href = "../../f77_src/barycentric_interp_1d/barycentric_interp_1d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/barycentric_interp_1d/barycentric_interp_1d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/barycentric_interp_1d/barycentric_interp_1d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/chebyshev_interp_1d/chebyshev_interp_1d.html">
CHEBYSHEV_INTERP_1D</a>,
a C++ library which
determines the combination of Chebyshev polynomials which
interpolates a set of data, so that p(x(i)) = y(i).
</p>
<p>
<a href = "../../m_src/lagrange_approx_1d/lagrange_approx_1d.html">
LAGRANGE_APPROX_1D</a>,
a MATLAB library which
defines and evaluates the Lagrange polynomial p(x) of degree m
which approximates a set of nd data points (x(i),y(i)).
</p>
<p>
<a href = "../../cpp_src/lagrange_interp_1d/lagrange_interp_1d.html">
LAGRANGE_INTERP_1D</a>,
a C++ library which
defines and evaluates the Lagrange polynomial p(x)
which interpolates a set of data, so that p(x(i)) = y(i).
</p>
<p>
<a href = "../../cpp_src/nearest_interp_1d/nearest_interp_1d.html">
NEAREST_INTERP_1D</a>,
a C++ library which
interpolates a set of data using a piecewise constant interpolant
defined by the nearest neighbor criterion.
</p>
<p>
<a href = "../../cpp_src/pwl_interp_1d/pwl_interp_1d.html">
PWL_INTERP_1D</a>,
a C++ library which
interpolates a set of data using a piecewise linear interpolant.
</p>
<p>
<a href = "../../cpp_src/r8lib/r8lib.html">
R8LIB</a>,
a C++ library which
contains many utility routines using double precision real (R8) arithmetic.
</p>
<p>
<a href = "../../cpp_src/rbf_interp_1d/rbf_interp_1d.html">
RBF_INTERP_1D</a>,
a C++ library which
defines and evaluates radial basis function (RBF) interpolants to 1D data.
</p>
<p>
<a href = "../../cpp_src/shepard_interp_1d/shepard_interp_1d.html">
SHEPARD_INTERP_1D</a>,
a C++ library which
defines and evaluates Shepard interpolants to 1D data,
based on inverse distance weighting.
</p>
<p>
<a href = "../../cpp_src/spline/spline.html">
SPLINE</a>,
a C++ library which
constructs and evaluates spline interpolants and approximants.
</p>
<p>
<a href = "../../cpp_src/test_interp_1d/test_interp_1d.html">
TEST_INTERP_1D</a>,
a C++ library which
defines test problems for interpolation of data y(x),
depending on a 2D argument.
</p>
<p>
<a href = "../../c_src/vandermonde_interp_1d/vandermonde_interp_1d.html">
VANDERMONDE_INTERP_1D</a>,
a C library which
finds a polynomial interpolant to data y(x) of a 1D argument,
by setting up and solving a linear system for the polynomial coefficients,
involving the Vandermonde matrix.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Kendall Atkinson,<br>
An Introduction to Numerical Analysis,<br>
Prentice Hall, 1989,<br>
ISBN: 0471624896,<br>
LC: QA297.A94.1989.
</li>
<li>
Jean-Paul Berrut, Lloyd Trefethen,<br>
Barycentric Lagrange Interpolation,<br>
SIAM Review,<br>
Volume 46, Number 3, September 2004, pages 501-517.
</li>
<li>
Philip Davis,<br>
Interpolation and Approximation,<br>
Dover, 1975,<br>
ISBN: 0-486-62495-1,<br>
LC: QA221.D33
</li>
<li>
David Kahaner, Cleve Moler, Steven Nash,<br>
Numerical Methods and Software,<br>
Prentice Hall, 1989,<br>
ISBN: 0-13-627258-4,<br>
LC: TA345.K34.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "barycentric_interp_1d.cpp">barycentric_interp_1d.cpp</a>, the source code.
</li>
<li>
<a href = "barycentric_interp_1d.hpp">barycentric_interp_1d.hpp</a>, the include file.
</li>
<li>
<a href = "barycentric_interp_1d.sh">barycentric_interp_1d.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "barycentric_interp_1d_prb.cpp">barycentric_interp_1d_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "barycentric_interp_1d_prb.sh">barycentric_interp_1d_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "barycentric_interp_1d_prb_output.txt">barycentric_interp_1d_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>LAGCHEBY1_INTERP_1D</b> evaluates the Lagrange Chebyshev 1 interpolant.
</li>
<li>
<b>LAGCHEBY2_INTERP_1D</b> evaluates the Lagrange Chebyshev 2 interpolant.
</li>
<li>
<b>LAGEVEN_VALUE_1D</b> evaluates the Lagrange evenly-spaced interpolant.
</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 14 October 2012.
</i>
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