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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 C++ library which
finds a polynomial interpolant to data by setting up and
solving a linear system involving the Vandermonde matrix,
creating graphics files for processing by gnuplot.
</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 CONDITION and TEST_INTERP libraries.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files 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 = "../../cpp_src/barycentric_interp_1d/barycentric_interp_1d.html">
BARYCENTRIC_INTERP_1D</a>,
a C++ 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 = "../../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 = "../../cpp_src/condition/condition.html">
CONDITION</a>,
a C++ library which
implements methods of computing or estimating the condition number of a matrix.
</p>
<p>
<a href = "../../cpp_src/divdif/divdif.html">
DIVDIF</a>,
a C++ library which
uses divided differences to compute the polynomial interpolant
to a given set of data.
</p>
<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/hermite/hermite.html">
HERMITE</a>,
a C++ library which
computes the Hermite interpolant, a polynomial that matches function values
and derivatives.
</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/qr_solve/qr_solve.html">
QR_SOLVE</a>,
a C++ library which
computes the least squares solution of a linear system A*x=b.
</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/test_interp.html">
TEST_INTERP</a>,
a C++ library which
defines a number of test problems for interpolation,
provided as a set of (x,y) data.
</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 = "../../cpp_src/vandermonde_approx_1d/vandermonde_approx_1d.html">
VANDERMONDE_APPROX_1D</a>,
a C++ library which
finds a polynomial approximant to data 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 = "../../cpp_src/vandermonde_interp_2d/vandermonde_interp_2d.html">
VANDERMONDE_INTERP_2D</a>,
a C++ 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.cpp">vandermonde_interp_1d.cpp</a>, the source code.
</li>
<li>
<a href = "vandermonde_interp_1d.hpp">vandermonde_interp_1d.hpp</a>, the include file.
</li>
<li>
<a href = "vandermonde_interp_1d.sh">vandermonde_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 = "vandermonde_interp_1d_prb.cpp">vandermonde_interp_1d_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "vandermonde_interp_1d_prb.sh">vandermonde_interp_1d_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "vandermonde_interp_1d_prb_output.txt">vandermonde_interp_1d_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
The test program makes data files that can be used by GNUPLOT to create graphics:
<ul>
<li>
<a href = "data1.txt">data1.txt</a>
</li>
<li>
<a href = "interp1.txt">interp1.txt</a>
</li>
<li>
<a href = "commands1.txt">commands1.txt</a>
</li>
<li>
<a href = "plot1.png">plot1.png</a>
</li>
<li>
<a href = "data2.txt">data2.txt</a>
</li>
<li>
<a href = "interp2.txt">interp2.txt</a>
</li>
<li>
<a href = "commands2.txt">commands2.txt</a>
</li>
<li>
<a href = "plot2.png">plot2.png</a>
</li>
<li>
<a href = "data3.txt">data3.txt</a>
</li>
<li>
<a href = "interp3.txt">interp3.txt</a>
</li>
<li>
<a href = "commands3.txt">commands3.txt</a>
</li>
<li>
<a href = "plot3.png">plot3.png</a>
</li>
<li>
<a href = "data4.txt">data4.txt</a>
</li>
<li>
<a href = "interp4.txt">interp4.txt</a>
</li>
<li>
<a href = "commands4.txt">commands4.txt</a>
</li>
<li>
<a href = "plot4.png">plot4.png</a>
</li>
<li>
<a href = "data5.txt">data5.txt</a>
</li>
<li>
<a href = "interp5.txt">interp5.txt</a>
</li>
<li>
<a href = "commands5.txt">commands5.txt</a>
</li>
<li>
<a href = "plot5.png">plot5.png</a>
</li>
<li>
<a href = "data6.txt">data6.txt</a>
</li>
<li>
<a href = "interp6.txt">interp6.txt</a>
</li>
<li>
<a href = "commands6.txt">commands6.txt</a>
</li>
<li>
<a href = "plot6.png">plot6.png</a>
</li>
<li>
<a href = "data7.txt">data7.txt</a>
</li>
<li>
<a href = "interp7.txt">interp7.txt</a>
</li>
<li>
<a href = "commands7.txt">commands7.txt</a>
</li>
<li>
<a href = "plot7.png">plot7.png</a>
</li>
<li>
<a href = "data8.txt">data8.txt</a>
</li>
<li>
<a href = "interp8.txt">interp8.txt</a>
</li>
<li>
<a href = "commands8.txt">commands8.txt</a>
</li>
<li>
<a href = "plot8.png">plot8.png</a>
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>VANDERMONDE_INTERP_1D_COEF</b> computes a 1D polynomial interpolant.
</li>
<li>
<b>VANDERMONDE_INTERP_1D_MATRIX</b> computes a Vandermonde 1D interpolation matrix.
</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 02 June 2013.
</i>
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