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<html>
<head>
<title>
VANDERMONDE - Accurate Solution of Vandermonde Systems
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
VANDERMONDE <br> Accurate Solution of Vandermonde Systems
</h1>
<hr>
<p>
<b>VANDERMONDE</b>
is a C++ library which
implements the Bjork-Pereyra algorithm for accurate solution of
linear systems involving the Vandermonde matrix.
</p>
<p>
A univariate NxN Vandermonde matrix is defined by a parameter vector
ALPHA of N distinct real values, and has the form:
<pre>
1 1 ... 1
alpha1 alpha2 ... alphan
alpha1^2 alpha2^2 ... alphan^2
... ... ... ...
alpha1^(n-1) alpha2^(n-1) ... alphan^(n-1)
</pre>
</p>
<p>
If p(x) is a polynomial of degree N-1, which is required to interpolate
a function f(x) at N distinct points ALPHA, then the coefficients
C of the polynomial can be found from the interpolation equations,
which can be written as a linear system involving a transposed
Vandermonde matrix:
<pre>
1 alpha1 alpha1^2 ... alpha1^(n-1) c0 f(alpha1)
1 alpha2 alpha2^2 ... alpha2^(n-1) c1 f(alpha2)
... ... ... ... ... ... = ...
1 alphan alphan^2 ... alphan^(n-1) c(n-1) f(alphan)
</pre>
</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</b> is available in
<a href = "../../c_src/vandermonde/vandermonde.html">a C version</a> and
<a href = "../../cpp_src/vandermonde/vandermonde.html">a C++ version</a> and
<a href = "../../f77_src/vandermonde/vandermonde.html">a FORTRAN77 version</a> and
<a href = "../../f_src/vandermonde/vandermonde.html">a FORTRAN90 version</a> and
<a href = "../../m_src/vandermonde/vandermonde.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/linplus/linplus.html">
LINPLUS</a>,
a C++ library which
carries out operations such as matrix-vector products,
matrix factorization, direct and iterative linear solvers
for matrices in a variety of formats, including banded, border-banded,
circulant, lower triangular, pentadiagonal, sparse, symmetric,
toeplitz, tridiagonal, upper triangular and vandermonde formats.
</p>
<p>
<a href = "../../cpp_src/quadrature_weights_vandermonde/quadrature_weights_vandermonde.html">
QUADRATURE_WEIGHTS_VANDERMONDE</a>,
a C++ library which
computes the weights of a quadrature rule using the Vandermonde
matrix, assuming that the points have been specified.
</p>
<p>
<a href = "../../cpp_src/test_mat/test_mat.html">
TEST_MAT</a>,
a C++ library which
defines test matrices for which some of the determinant, eigenvalues,
inverse, null vectors, P*L*U factorization or linear system solution
are already known, including the Vandermonde matrix.
</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_approx_2d/vandermonde_approx_2d.html">
VANDERMONDE_APPROX_2D</a>,
a C++ library which
finds a polynomial approximant p(x,y) to data z(x,y) of a 2D 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_1d/vandermonde_interp_1d.html">
VANDERMONDE_INTERP_1D</a>,
a C++ 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>
<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>
Ake Bjorck, Victor Pereyra,<br>
Solution of Vandermonde Systems of Equations,<br>
Mathematics of Computation,<br>
Volume 24, Number 112, October 1970, pages 893-903.
</li>
<li>
Gustavo Galimberti, Victor Pereyra,<br>
Numerical Differentiation and the Solution of Multidimensional Vandermonde
Systems,<br>
Mathematics of Computation,<br>
Volume 24, Number 110, April 1970, pages 357-364.
</li>
<li>
Nicholas Higham,<br>
Stability analysis of algorithms for solving confluent
Vandermonde-like systems,<br>
SIAM Journal on Matrix Analysis and Applications,<br>
Volume 11, 1990, pages 23-41.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "vandermonde.cpp">vandermonde.cpp</a>, the source code.
</li>
<li>
<a href = "vandermonde.hpp">vandermonde.hpp</a>, the include file.
</li>
<li>
<a href = "vandermonde.sh">vandermonde.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_prb.cpp">vandermonde_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "vandermonde_prb.sh">vandermonde_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "vandermonde_prb_output.txt">vandermonde_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>BIVAND1</b> returns a bidimensional Vandermonde1 matrix.
</li>
<li>
<b>DVAND</b> solves a Vandermonde system A' * x = b.
</li>
<li>
<b>DVANDPRG</b> solves a Vandermonde system A' * x = f progressively.
</li>
<li>
<b>PVAND</b> solves a Vandermonde system A * x = b.
</li>
<li>
<b>PVANDPRG</b> solves a Vandermonde system A * x = f progressively.
</li>
<li>
<b>R8MAT_MTV_NEW</b> multiplies a transposed matrix times a vector.
</li>
<li>
<b>R8MAT_MV_NEW</b> multiplies a matrix times a vector.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</li>
<li>
<b>R8VEC_COPY_NEW</b> copies an R8VEC to a new R8VEC.
</li>
<li>
<b>R8VEC_PRINT</b> prints an R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01_NEW</b> returns a new unit pseudorandom R8VEC.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>VAND1</b> returns the Vandermonde1 matrix A with 1's on the first row.
</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 18 April 2014.
</i>
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