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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 MATLAB 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 = "../../m_src/chebyshev_interp_1d/chebyshev_interp_1d.html">
CHEBYSHEV_INTERP_1D</a>,
a MATLAB 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 = "../../m_src/lagrange_basis_display/lagrange_basis_display.html">
LAGRANGE_BASIS_DISPLAY</a>,
a MATLAB library which
displays the basis functions associated with a given set of nodes used
with the Lagrange interpolation scheme.
</p>
<p>
<a href = "../../m_src/lagrange_interp_1d/lagrange_interp_1d.html">
LAGRANGE_INTERP_1D</a>,
a MATLAB 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 = "../../m_src/nearest_interp_1d/nearest_interp_1d.html">
NEAREST_INTERP_1D</a>,
a MATLAB library which
interpolates a set of data using a piecewise constant interpolant
defined by the nearest neighbor criterion.
</p>
<p>
<a href = "../../m_src/pwl_interp_1d/pwl_interp_1d.html">
PWL_INTERP_1D</a>,
a MATLAB library which
interpolates a set of data using a piecewise linear interpolant.
</p>
<p>
<a href = "../../m_src/r8lib/r8lib.html">
R8LIB</a>,
a MATLAB library which
contains many utility routines using double precision real (R8) arithmetic.
</p>
<p>
<a href = "../../m_src/rbf_interp_1d/rbf_interp_1d.html">
RBF_INTERP_1D</a>,
a MATLAB library which
defines and evaluates radial basis function (RBF) interpolants to 1D data.
</p>
<p>
<a href = "../../m_src/shepard_interp_1d/shepard_interp_1d.html">
SHEPARD_INTERP_1D</a>,
a MATLAB library which
defines and evaluates Shepard interpolants to 1D data,
which are based on inverse distance weighting.
</p>
<p>
<a href = "../../m_src/spline/spline.html">
SPLINE</a>,
a MATLAB library which
constructs and evaluates spline interpolants and approximants.
</p>
<p>
<a href = "../../m_src/test_interp/test_interp.html">
TEST_INTERP</a>,
a MATLAB library which
defines a number of test problems for interpolation,
provided as a set of (x,y) data.
</p>
<p>
<a href = "../../m_src/test_interp_1d/test_interp_1d.html">
TEST_INTERP_1D</a>,
a MATLAB library which
defines test problems for interpolation of data y(x),
depending on a 2D argument.
</p>
<p>
<a href = "../../m_src/vandermonde_interp_1d/vandermonde_interp_1d.html">
VANDERMONDE_INTERP_1D</a>,
a MATLAB library which
finds a polynomial interpolant to a function of 1D data
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 = "lageven_interp_1d.m">lageven_interp_1d.m</a>,
evaluates the barycentric Lagrange interpolant in cases where
the data is evaluated at evenly-spaced midpoints.
</li>
<li>
<a href = "lagcheby1_interp_1d.m">lagcheby1_interp_1d.m</a>,
evaluates the barycentric Lagrange interpolant in cases where
the data is evaluated at the Chebyshev type 1 points.
</li>
<li>
<a href = "lagcheby2_interp_1d.m">lagcheby2_interp_1d.m</a>,
evaluates the barycentric Lagrange interpolant in cases where
the data is evaluated at the Chebyshev type 2 points.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
The test code requires the <b>test_interp_1d</b> library. If this library is
available in a separate folder at the same "level",
then a Matlab command such as "addpath ( '../test_interp_1d')" will make it
accessible for a run of the test program.
<ul>
<li>
<a href = "barycentric_interp_1d_test.m">barycentric_interp_1d_test.m</a>, calls all the tests;
</li>
<li>
<a href = "barycentric_interp_1d_test_output.txt">barycentric_interp_1d_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "barycentric_interp_1d_test01.m">barycentric_interp_1d_test01.m</a>,
samples a function at Chebyshev Type 1 points and determines and plots
the barycentric Lagrange polynomial interpolant.
</li>
<li>
<a href = "barycentric_interp_1d_test02.m">barycentric_interp_1d_test02.m</a>,
samples a function at Chebyshev Type 2 points and determines and plots
the barycentric Lagrange polynomial interpolant.
</li>
</ul>
</p>
<p>
<b>barycentric_interp_1d_test01()</b>
samples a function at Chebyshev Type 1 points and determines and plots
the barycentric Lagrange polynomial interpolant.
<ul>
<li>
<a href = "p01_lagcheby1_04.png">p01_lagcheby1_04.png</a>,
a plot of the interpolant to 4 Chebyshev 1 points for problem p01;
</li>
<li>
<a href = "p01_lagcheby1_08.png">p01_lagcheby1_08.png</a>,
a plot of the interpolant to 8 Chebyshev 1 points for problem p01;
</li>
<li>
<a href = "p01_lagcheby1_16.png">p01_lagcheby1_16.png</a>,
a plot of the interpolant to 16 Chebyshev 1 points for problem p01;
</li>
<li>
<a href = "p01_lagcheby1_32.png">p01_lagcheby1_32.png</a>,
a plot of the interpolant to 32 Chebyshev 1 points for problem p01;
</li>
<li>
<a href = "p01_lagcheby1_64.png">p01_lagcheby1_64.png</a>,
a plot of the interpolant to 64 Chebyshev 1 points for problem p01;
</li>
<li>
<a href = "p02_lagcheby1_04.png">p02_lagcheby1_04.png</a>,
a plot of the interpolant to 4 Chebyshev 1 points for problem p02;
</li>
<li>
<a href = "p02_lagcheby1_08.png">p02_lagcheby1_08.png</a>,
a plot of the interpolant to 8 Chebyshev 1 points for problem p02;
</li>
<li>
<a href = "p02_lagcheby1_16.png">p02_lagcheby1_16.png</a>,
a plot of the interpolant to 16 Chebyshev 1 points for problem p02;
</li>
<li>
<a href = "p02_lagcheby1_32.png">p02_lagcheby1_32.png</a>,
a plot of the interpolant to 32 Chebyshev 1 points for problem p02;
</li>
<li>
<a href = "p02_lagcheby1_64.png">p02_lagcheby1_64.png</a>,
a plot of the interpolant to 64 Chebyshev 1 points for problem p02;
</li>
<li>
<a href = "p03_lagcheby1_04.png">p03_lagcheby1_04.png</a>,
a plot of the interpolant to 4 Chebyshev 1 points for problem p03;
</li>
<li>
<a href = "p03_lagcheby1_08.png">p03_lagcheby1_08.png</a>,
a plot of the interpolant to 8 Chebyshev 1 points for problem p03;
</li>
<li>
<a href = "p03_lagcheby1_16.png">p03_lagcheby1_16.png</a>,
a plot of the interpolant to 16 Chebyshev 1 points for problem p03;
</li>
<li>
<a href = "p03_lagcheby1_32.png">p03_lagcheby1_32.png</a>,
a plot of the interpolant to 32 Chebyshev 1 points for problem p03;
</li>
<li>
<a href = "p03_lagcheby1_64.png">p03_lagcheby1_64.png</a>,
a plot of the interpolant to 64 Chebyshev 1 points for problem p03;
</li>
<li>
<a href = "p04_lagcheby1_04.png">p04_lagcheby1_04.png</a>,
a plot of the interpolant to 4 Chebyshev 1 points for problem p04;
</li>
<li>
<a href = "p04_lagcheby1_08.png">p04_lagcheby1_08.png</a>,
a plot of the interpolant to 8 Chebyshev 1 points for problem p04;
</li>
<li>
<a href = "p04_lagcheby1_16.png">p04_lagcheby1_16.png</a>,
a plot of the interpolant to 16 Chebyshev 1 points for problem p04;
</li>
<li>
<a href = "p04_lagcheby1_32.png">p04_lagcheby1_32.png</a>,
a plot of the interpolant to 32 Chebyshev 1 points for problem p04;
</li>
<li>
<a href = "p04_lagcheby1_64.png">p04_lagcheby1_64.png</a>,
a plot of the interpolant to 64 Chebyshev 1 points for problem p04;
</li>
<li>
<a href = "p05_lagcheby1_04.png">p05_lagcheby1_04.png</a>,
a plot of the interpolant to 4 Chebyshev 1 points for problem p05;
</li>
<li>
<a href = "p05_lagcheby1_08.png">p05_lagcheby1_08.png</a>,
a plot of the interpolant to 8 Chebyshev 1 points for problem p05;
</li>
<li>
<a href = "p05_lagcheby1_16.png">p05_lagcheby1_16.png</a>,
a plot of the interpolant to 16 Chebyshev 1 points for problem p05;
</li>
<li>
<a href = "p05_lagcheby1_32.png">p05_lagcheby1_32.png</a>,
a plot of the interpolant to 32 Chebyshev 1 points for problem p05;
</li>
<li>
<a href = "p05_lagcheby1_64.png">p05_lagcheby1_64.png</a>,
a plot of the interpolant to 64 Chebyshev 1 points for problem p05;
</li>
<li>
<a href = "p06_lagcheby1_04.png">p06_lagcheby1_04.png</a>,
a plot of the interpolant to 4 Chebyshev 1 points for problem p06;
</li>
<li>
<a href = "p06_lagcheby1_08.png">p06_lagcheby1_08.png</a>,
a plot of the interpolant to 8 Chebyshev 1 points for problem p06;
</li>
<li>
<a href = "p06_lagcheby1_16.png">p06_lagcheby1_16.png</a>,
a plot of the interpolant to 16 Chebyshev 1 points for problem p06;
</li>
<li>
<a href = "p06_lagcheby1_32.png">p06_lagcheby1_32.png</a>,
a plot of the interpolant to 32 Chebyshev 1 points for problem p06;
</li>
<li>
<a href = "p06_lagcheby1_64.png">p06_lagcheby1_64.png</a>,
a plot of the interpolant to 64 Chebyshev 1 points for problem p06;
</li>
<li>
<a href = "p07_lagcheby1_04.png">p07_lagcheby1_04.png</a>,
a plot of the interpolant to 4 Chebyshev 1 points for problem p07;
</li>
<li>
<a href = "p07_lagcheby1_08.png">p07_lagcheby1_08.png</a>,
a plot of the interpolant to 8 Chebyshev 1 points for problem p07;
</li>
<li>
<a href = "p07_lagcheby1_16.png">p07_lagcheby1_16.png</a>,
a plot of the interpolant to 16 Chebyshev 1 points for problem p07;
</li>
<li>
<a href = "p07_lagcheby1_32.png">p07_lagcheby1_32.png</a>,
a plot of the interpolant to 32 Chebyshev 1 points for problem p07;
</li>
<li>
<a href = "p07_lagcheby1_64.png">p07_lagcheby1_64.png</a>,
a plot of the interpolant to 64 Chebyshev 1 points for problem p07;
</li>
<li>
<a href = "p08_lagcheby1_04.png">p08_lagcheby1_04.png</a>,
a plot of the interpolant to 4 Chebyshev 1 points for problem p08;
</li>
<li>
<a href = "p08_lagcheby1_08.png">p08_lagcheby1_08.png</a>,
a plot of the interpolant to 8 Chebyshev 1 points for problem p08;
</li>
<li>
<a href = "p08_lagcheby1_16.png">p08_lagcheby1_16.png</a>,
a plot of the interpolant to 16 Chebyshev 1 points for problem p08;
</li>
<li>
<a href = "p08_lagcheby1_32.png">p08_lagcheby1_32.png</a>,
a plot of the interpolant to 32 Chebyshev 1 points for problem p08;
</li>
<li>
<a href = "p08_lagcheby1_64.png">p08_lagcheby1_64.png</a>,
a plot of the interpolant to 64 Chebyshev 1 points for problem p08;
</li>
</ul>
</p>
<p>
<b>barycentric_interp_1d_test02()</b>
samples a function at Chebyshev Type 2 points and determines and plots
the barycentric Lagrange polynomial interpolant.
<ul>
<li>
<a href = "p01_lagcheby2_04.png">p01_lagcheby2_04.png</a>,
a plot of the interpolant to 4 Chebyshev 2 points for problem p01;
</li>
<li>
<a href = "p01_lagcheby2_08.png">p01_lagcheby2_08.png</a>,
a plot of the interpolant to 8 Chebyshev 2 points for problem p01;
</li>
<li>
<a href = "p01_lagcheby2_16.png">p01_lagcheby2_16.png</a>,
a plot of the interpolant to 16 Chebyshev 2 points for problem p01;
</li>
<li>
<a href = "p01_lagcheby2_32.png">p01_lagcheby2_32.png</a>,
a plot of the interpolant to 32 Chebyshev 2 points for problem p01;
</li>
<li>
<a href = "p01_lagcheby2_64.png">p01_lagcheby2_64.png</a>,
a plot of the interpolant to 64 Chebyshev 2 points for problem p01;
</li>
<li>
<a href = "p02_lagcheby2_04.png">p02_lagcheby2_04.png</a>,
a plot of the interpolant to 4 Chebyshev 2 points for problem p02;
</li>
<li>
<a href = "p02_lagcheby2_08.png">p02_lagcheby2_08.png</a>,
a plot of the interpolant to 8 Chebyshev 2 points for problem p02;
</li>
<li>
<a href = "p02_lagcheby2_16.png">p02_lagcheby2_16.png</a>,
a plot of the interpolant to 16 Chebyshev 2 points for problem p02;
</li>
<li>
<a href = "p02_lagcheby2_32.png">p02_lagcheby2_32.png</a>,
a plot of the interpolant to 32 Chebyshev 2 points for problem p02;
</li>
<li>
<a href = "p02_lagcheby2_64.png">p02_lagcheby2_64.png</a>,
a plot of the interpolant to 64 Chebyshev 2 points for problem p02;
</li>
<li>
<a href = "p03_lagcheby2_04.png">p03_lagcheby2_04.png</a>,
a plot of the interpolant to 4 Chebyshev 2 points for problem p03;
</li>
<li>
<a href = "p03_lagcheby2_08.png">p03_lagcheby2_08.png</a>,
a plot of the interpolant to 8 Chebyshev 2 points for problem p03;
</li>
<li>
<a href = "p03_lagcheby2_16.png">p03_lagcheby2_16.png</a>,
a plot of the interpolant to 16 Chebyshev 2 points for problem p03;
</li>
<li>
<a href = "p03_lagcheby2_32.png">p03_lagcheby2_32.png</a>,
a plot of the interpolant to 32 Chebyshev 2 points for problem p03;
</li>
<li>
<a href = "p03_lagcheby2_64.png">p03_lagcheby2_64.png</a>,
a plot of the interpolant to 64 Chebyshev 2 points for problem p03;
</li>
<li>
<a href = "p04_lagcheby2_04.png">p04_lagcheby2_04.png</a>,
a plot of the interpolant to 4 Chebyshev 2 points for problem p04;
</li>
<li>
<a href = "p04_lagcheby2_08.png">p04_lagcheby2_08.png</a>,
a plot of the interpolant to 8 Chebyshev 2 points for problem p04;
</li>
<li>
<a href = "p04_lagcheby2_16.png">p04_lagcheby2_16.png</a>,
a plot of the interpolant to 16 Chebyshev 2 points for problem p04;
</li>
<li>
<a href = "p04_lagcheby2_32.png">p04_lagcheby2_32.png</a>,
a plot of the interpolant to 32 Chebyshev 2 points for problem p04;
</li>
<li>
<a href = "p04_lagcheby2_64.png">p04_lagcheby2_64.png</a>,
a plot of the interpolant to 64 Chebyshev 2 points for problem p04;
</li>
<li>
<a href = "p05_lagcheby2_04.png">p05_lagcheby2_04.png</a>,
a plot of the interpolant to 4 Chebyshev 2 points for problem p05;
</li>
<li>
<a href = "p05_lagcheby2_08.png">p05_lagcheby2_08.png</a>,
a plot of the interpolant to 8 Chebyshev 2 points for problem p05;
</li>
<li>
<a href = "p05_lagcheby2_16.png">p05_lagcheby2_16.png</a>,
a plot of the interpolant to 16 Chebyshev 2 points for problem p05;
</li>
<li>
<a href = "p05_lagcheby2_32.png">p05_lagcheby2_32.png</a>,
a plot of the interpolant to 32 Chebyshev 2 points for problem p05;
</li>
<li>
<a href = "p05_lagcheby2_64.png">p05_lagcheby2_64.png</a>,
a plot of the interpolant to 64 Chebyshev 2 points for problem p05;
</li>
<li>
<a href = "p06_lagcheby2_04.png">p06_lagcheby2_04.png</a>,
a plot of the interpolant to 4 Chebyshev 2 points for problem p06;
</li>
<li>
<a href = "p06_lagcheby2_08.png">p06_lagcheby2_08.png</a>,
a plot of the interpolant to 8 Chebyshev 2 points for problem p06;
</li>
<li>
<a href = "p06_lagcheby2_16.png">p06_lagcheby2_16.png</a>,
a plot of the interpolant to 16 Chebyshev 2 points for problem p06;
</li>
<li>
<a href = "p06_lagcheby2_32.png">p06_lagcheby2_32.png</a>,
a plot of the interpolant to 32 Chebyshev 2 points for problem p06;
</li>
<li>
<a href = "p06_lagcheby2_64.png">p06_lagcheby2_64.png</a>,
a plot of the interpolant to 64 Chebyshev 2 points for problem p06;
</li>
<li>
<a href = "p07_lagcheby2_04.png">p07_lagcheby2_04.png</a>,
a plot of the interpolant to 4 Chebyshev 2 points for problem p07;
</li>
<li>
<a href = "p07_lagcheby2_08.png">p07_lagcheby2_08.png</a>,
a plot of the interpolant to 8 Chebyshev 2 points for problem p07;
</li>
<li>
<a href = "p07_lagcheby2_16.png">p07_lagcheby2_16.png</a>,
a plot of the interpolant to 16 Chebyshev 2 points for problem p07;
</li>
<li>
<a href = "p07_lagcheby2_32.png">p07_lagcheby2_32.png</a>,
a plot of the interpolant to 32 Chebyshev 2 points for problem p07;
</li>
<li>
<a href = "p07_lagcheby2_64.png">p07_lagcheby2_64.png</a>,
a plot of the interpolant to 64 Chebyshev 2 points for problem p07;
</li>
<li>
<a href = "p08_lagcheby2_04.png">p08_lagcheby2_04.png</a>,
a plot of the interpolant to 4 Chebyshev 2 points for problem p08;
</li>
<li>
<a href = "p08_lagcheby2_08.png">p08_lagcheby2_08.png</a>,
a plot of the interpolant to 8 Chebyshev 2 points for problem p08;
</li>
<li>
<a href = "p08_lagcheby2_16.png">p08_lagcheby2_16.png</a>,
a plot of the interpolant to 16 Chebyshev 2 points for problem p08;
</li>
<li>
<a href = "p08_lagcheby2_32.png">p08_lagcheby2_32.png</a>,
a plot of the interpolant to 32 Chebyshev 2 points for problem p08;
</li>
<li>
<a href = "p08_lagcheby2_64.png">p08_lagcheby2_64.png</a>,
a plot of the interpolant to 64 Chebyshev 2 points for problem p08;
</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 modified on 19 August 2012.
</i>
<!-- John Burkardt -->
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