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<html>
<head>
<title>
VANDERMONDE_APPROX_2D - Data Approximation with Polynomials using the Vandermonde Matrix
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
VANDERMONDE_APPROX_2D <br> Data Approximation with Polynomials using the Vandermonde Matrix
</h1>
<hr>
<p>
<b>VANDERMONDE_APPROX_2D</b>
is a C++ library which
finds P(X,Y), a polynomial approximant to Z which depends on two independent
variables X and Y, by setting up and solving an overdetermined linear system
involving the Vandermonde matrix.
</p>
<p>
This software is primarily intended as an illustration of the problems
that can occur when the approximation problem is naively formulated
using the Vandermonde matrix. Unless the data points are well separated,
and the degree of the polynomial is low, the linear system will become
very difficult to store and solve accurately, because the monomials
used as basis vectors by the Vandermonde approach become indistinguishable.
</p>
<p>
<b>VANDERMONDE_INTERP_2D</b> needs access to the QR_SOLVE and R8LIB libraries.
The test code also needs access to the TEST_INTERP_2D 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_APPROX_2D</b> is available in
<a href = "../../c_src/vandermonde_approx_2d/vandermonde_approx_2d.html">a C version</a> and
<a href = "../../cpp_src/vandermonde_approx_2d/vandermonde_approx_2d.html">a C++ version</a> and
<a href = "../../f77_src/vandermonde_approx_2d/vandermonde_approx_2d.html">a FORTRAN77 version</a> and
<a href = "../../f_src/vandermonde_approx_2d/vandermonde_approx_2d.html">a FORTRAN90 version</a> and
<a href = "../../m_src/vandermonde_approx_2d/vandermonde_approx_2d.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<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/test_interp_2d/test_interp_2d.html">
TEST_INTERP_2D</a>,
a C++ library which
defines test problems for interpolation of data (x,y,z(x,y)),
with the data points (x,y) scattered irregularly.
</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 y(x) 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_approx_2d.cpp">vandermonde_approx_2d.cpp</a>, the source code.
</li>
<li>
<a href = "vandermonde_approx_2d.hpp">vandermonde_approx_2d.hpp</a>, the include file.
</li>
<li>
<a href = "vandermonde_approx_2d.sh">vandermonde_approx_2d.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_approx_2d_prb.cpp">vandermonde_approx_2d_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "vandermonde_approx_2d_prb.sh">vandermonde_approx_2d_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "vandermonde_approx_2d_prb_output.txt">vandermonde_approx_2d_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>TRIANGLE_NUM</b> returns the N-th triangular number.
</li>
<li>
<b>VANDERMONDE_APPROX_2D_COEF</b> computes a 2D polynomial approximant.
</li>
<li>
<b>VANDERMONDE_APPROX_2D_MATRIX</b> computes a Vandermonde 2D approximation 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 12 October 2012.
</i>
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