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<html>
<head>
<title>
VANDERMONDE_INTERP_1D - Polynomial Interpolation with the Vandermonde Matrix
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
VANDERMONDE_INTERP_1D <br> Polynomial Interpolation with the Vandermonde Matrix
</h1>
<hr>
<p>
<b>VANDERMONDE_INTERP_1D</b>
is a MATLAB library which
finds a polynomial interpolant to data by setting up and
solving a linear system involving the Vandermonde matrix.
</p>
<p>
This software is primarily intended as an illustration of the problems
that can occur when the interpolation problem is naively formulated
using the Vandermonde matrix. If the underlying interpolating basis
is the usual family of monomials, then the Vandermonde matrix will
very quickly become ill-conditioned for almost any set of nodes.
</p>
<p>
If the nodes can be selected, this can provide a small amount of improvement,
but, if a polynomial interpolant is desired, a better strategy is to change
the basis, which is what is done with the Lagrange interpolation method,
in which case, essentially, the linear system to be solved becomes the
identity matrix.
</p>
<p>
<b>VANDERMONDE_INTERP_1D</b> needs access to the QR_SOLVE and R8LIB libraries.
The test code also needs access to the TEST_INTERP 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>VANDERMONDE_INTERP_1D</b> is available in
<a href = "../../c_src/vandermonde_interp_1d/vandermonde_interp_1d.html">a C version</a> and
<a href = "../../cpp_src/vandermonde_interp_1d/vandermonde_interp_1d.html">a C++ version</a> and
<a href = "../../f77_src/vandermonde_interp_1d/vandermonde_interp_1d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/vandermonde_interp_1d/vandermonde_interp_1d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/vandermonde_interp_1d/vandermonde_interp_1d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/barycentric_interp_1d/barycentric_interp_1d.html">
BARYCENTRIC_INTERP_1D</a>,
a MATLAB library which
defines and evaluates the barycentric Lagrange polynomial p(x)
which interpolates a set of data, so that p(x(i)) = y(i).
The barycentric approach means that very high degree polynomials can
safely be used.
</p>
<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/divdif/divdif.html">
DIVDIF</a>,
a MATLAB library which
uses divided differences to compute the polynomial interpolant
to a given set of data.
</p>
<p>
<a href = "../../m_src/hermite/hermite.html">
HERMITE</a>,
a MATLAB library which
computes the Hermite interpolant, a polynomial that matches function values
and derivatives.
</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_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_approx_1d/vandermonde_approx_1d.html">
VANDERMONDE_APPROX_1D</a>,
a MATLAB library which
finds a polynomial approximant to data z(x,y) of a 1D argument
by setting up and solving an overdetermined linear system for the polynomial coefficients,
involving the Vandermonde matrix.
</p>
<p>
<a href = "../../m_src/vandermonde_interp_2d/vandermonde_interp_2d.html">
VANDERMONDE_INTERP_2D</a>,
a MATLAB library which
finds a polynomial interpolant to data z(x,y) of a 2D 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>
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 = "vandermonde_interp_1d_coef.m">vandermonde_interp_1d_coef.m</a>,
solves the Vandermonde system of equations for the coefficients of the
polynomial that interpolates a given set of (x,y) data.
</li>
<li>
<a href = "vandermonde_interp_1d_matrix.m">vandermonde_interp_1d_matrix.m</a>,
returns the Vandermonde matrix associated with a given set of nodes x.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
The test code requires the <b>test_interp</b> library as well. If this library is
available in a separate folder at the same "level" as this library,
then a Matlab command such as "addpath ( '../test_interp')" will make that library
accessible for a run of the test program.
<ul>
<li>
<a href = "vandermonde_interp_1d_test.m">vandermonde_interp_1d_test.m</a>, calls all the tests;
</li>
<li>
<a href = "vandermonde_interp_1d_test_output.txt">vandermonde_interp_1d_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "vandermonde_interp_1d_test01.m">vandermonde_interp_1d_test01.m</a>,
computes the interpolant for a given problem.
</li>
</ul>
</p>
<p>
The code generates some plots of the data and approximants.
<ul>
<li>
<a href = "p01_data.png">p01_data.png</a>,
a plot of the data and piecewise linear interpolant for problem p01;
</li>
<li>
<a href = "p01_poly.png">p01_poly.png</a>,
a plot of the polynomial interpolant for problem p01;
</li>
<li>
<a href = "p02_data.png">p02_data.png</a>,
a plot of the data and piecewise linear interpolant for problem p02;
</li>
<li>
<a href = "p02_poly.png">p02_poly.png</a>,
a plot of the polynomial interpolant for problem p02;
</li>
<li>
<a href = "p03_data.png">p03_data.png</a>,
a plot of the data and piecewise linear interpolant for problem p03;
</li>
<li>
<a href = "p03_poly.png">p03_poly.png</a>,
a plot of the polynomial interpolant for problem p03;
</li>
<li>
<a href = "p04_data.png">p04_data.png</a>,
a plot of the data and piecewise linear interpolant for problem p04;
</li>
<li>
<a href = "p04_poly.png">p04_poly.png</a>,
a plot of the polynomial interpolant for problem p04;
</li>
<li>
<a href = "p05_data.png">p05_data.png</a>,
a plot of the data and piecewise linear interpolant for problem p05;
</li>
<li>
<a href = "p05_poly.png">p05_poly.png</a>,
a plot of the polynomial interpolant for problem p05;
</li>
<li>
<a href = "p06_data.png">p06_data.png</a>,
a plot of the data and piecewise linear interpolant for problem p06;
</li>
<li>
<a href = "p06_poly.png">p06_poly.png</a>,
a plot of the polynomial interpolant for problem p06;
</li>
<li>
<a href = "p07_data.png">p07_data.png</a>,
a plot of the data and piecewise linear interpolant for problem p07;
</li>
<li>
<a href = "p07_poly.png">p07_poly.png</a>,
a plot of the polynomial interpolant for problem p07;
</li>
<li>
<a href = "p08_data.png">p08_data.png</a>,
a plot of the data and piecewise linear interpolant for problem p08;
</li>
<li>
<a href = "p08_poly.png">p08_poly.png</a>,
a plot of the polynomial interpolant 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 29 July 2012.
</i>
<!-- John Burkardt -->
</body>
</html>