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<html>
<head>
<title>
SVD_DEMO - Demonstration of the Singular Value Decomposition
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SVD_DEMO <br> Demonstration of the Singular Value Decomposition
</h1>
<hr>
<p>
<b>SVD_DEMO</b>
is a C++ program which
demonstrates the computation of the singular value decomposition
and a few of its properties.
</p>
<p>
The singular value decomposition has uses in solving
overdetermined or underdetermined linear systems,
linear least squares problems, data compression,
the pseudoinverse matrix,
reduced order modeling, and
the accurate computation of matrix rank and null space.
</p>
<p>
The singular value decomposition of an M by N rectangular matrix A has
the form
<pre>
A(mxn) = U(mxm) * S(mxn) * V'(nxn)
</pre>
where
<ul>
<li>
U is an orthogonal matrix, whose columns are the left singular vectors;
</li>
<li>
S is a diagonal matrix, whose min(m,n) diagonal entries are the singular values;
</li>
<li>
V is an orthogonal matrix, whose columns are the right singular vectors;
</li>
</ul>
Note that the transpose of V is used in the decomposition, and that the diagonal matrix
S is typically stored as a vector.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
<blockquote>
<b>svd_demo</b> <i>m</i> <i>n</i> <i>seed</i>
</blockquote>
where
<ul>
<li>
<i>m</i> is the number of rows in the random matrix;
</li>
<li>
<i>n</i> is the number of columns in the random matrix;
</li>
<li>
<i>seed</i> is an optional seed for the random number generator.
</li>
</ul>
</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>SVD_DEMO</b> is available in
<a href = "../../c_src/svd_demo/svd_demo.html">a C version</a> and
<a href = "../../cpp_src/svd_demo/svd_demo.html">a C++ version</a> and
<a href = "../../f77_src/svd_demo/svd_demo.html">a FORTRAN77 version</a> and
<a href = "../../f_src/svd_demo/svd_demo.html">a FORTRAN90 version</a> and
<a href = "../../m_src/svd_demo/svd_demo.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../datasets/fingerprints/fingerprints.html">
FINGERPRINTS</a>,
a dataset directory which
contains a few images of fingerprints.
</p>
<p>
<a href = "../../f_src/lapack_examples/lapack_examples.html">
LAPACK_EXAMPLES</a>,
a FORTRAN90 program which
demonstrates the use of the LAPACK linear algebra library.
</p>
<p>
<a href = "../../cpp_src/linpack/linpack.html">
LINPACK</a>,
a C++ library which
includes routines to carry out the singular value
decomposition.
</p>
<p>
<a href = "../../cpp_src/svd_basis/svd_basis.html">
SVD_BASIS</a>,
a C++ program which
computes a reduced basis for a collection of data vectors using the SVD.
</p>
<p>
<a href = "../../cpp_src/svd_snowfall/svd_snowfall.html">
SVD_SNOWFALL</a>,
a C++ library which
reads a file containing historical snowfall data and
analyzes the data with the Singular Value Decomposition (SVD),
and plots created by GNUPLOT.
</p>
<p>
<a href = "../../cpp_src/svd_truncated/svd_truncated.html">
SVD_TRUNCATED</a>,
a C++ program which
demonstrates the computation of the reduced or truncated
Singular Value Decomposition (SVD) that is useful for cases when
one dimension of the matrix is much smaller than the other.
</p>
<p>
<a href = "../../f77_src/toms358/toms358.html">
TOMS358</a>,
a FORTRAN77 routine which
computes the singular value decomposition for a complex matrix.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Edward Anderson, Zhaojun Bai, Christian Bischof, Susan Blackford,
James Demmel, Jack Dongarra, Jeremy Du Croz, Anne Greenbaum,
Sven Hammarling, Alan McKenney, Danny Sorensen,<br>
LAPACK User's Guide,<br>
Third Edition,<br>
SIAM, 1999,<br>
ISBN: 0898714478,<br>
LC: QA76.73.F25L36
</li>
<li>
Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart,<br>
LINPACK User's Guide,<br>
SIAM, 1979,<br>
ISBN13: 978-0-898711-72-1,<br>
LC: QA214.L56.
</li>
<li>
Gene Golub, Charles VanLoan,<br>
Matrix Computations,
Third Edition,<br>
Johns Hopkins, 1996,<br>
ISBN: 0-8018-4513-X,<br>
LC: QA188.G65.
</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>
<li>
Lloyd Trefethen, David Bau,<br>
Numerical Linear Algebra,<br>
SIAM, 1997,<br>
ISBN: 0-89871-361-7,<br>
LC: QA184.T74.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "svd_demo.cpp">svd_demo.cpp</a>, the source code.
</li>
<li>
<a href = "svd_demo.sh">svd_demo.sh</a>,
commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "svd_demo_5_3_output.txt">svd_demo_5_3_output.txt</a>,
the output from the command "svd_demo 5 3 123456789".
</li>
<li>
<a href = "svd_demo_5_5_output.txt">svd_demo_5_5_output.txt</a>,
the output from the command "svd_demo 5 5 123456789".
</li>
<li>
<a href = "svd_demo_5_8_output.txt">svd_demo_5_8_output.txt</a>,
the output from the command "svd_demo 5 8 123456789".
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for SVD_DEMO.
</li>
<li>
<b>GET_SEED</b> returns a random seed for the random number generator.
</li>
<li>
<b>GET_SVD_LINPACK</b> gets the SVD of a matrix using a call to LINPACK.
</li>
<li>
<b>PSEUDO_INVERSE</b> computes the pseudoinverse.
</li>
<li>
<b>PSEUDO_LINEAR_SOLVE_TEST</b> uses the pseudoinverse for linear systems.
</li>
<li>
<b>PSEUDO_PRODUCT_TEST</b> examines pseudoinverse products.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to a double precision value.
</li>
<li>
<b>R8MAT_DIF_FRO</b> returns the Frobenius norm of the difference of R8MAT's.
</li>
<li>
<b>R8MAT_NORM_FRO</b> returns the Frobenius norm of a R8MAT.
</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_NORM_L2</b> returns the L2 norm of an R8VEC.
</li>
<li>
<b>R8MAT_UNIFORM_01</b> returns a unit pseudorandom R8MAT.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>RANK_ONE_PRINT_TEST</b> prints the sums of rank one matrices.
</li>
<li>
<b>RANK_ONE_TEST</b> compares A to the sum of rank one matrices.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>SVD_PRODUCT_TEST</b> tests that A = U * S * V'.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</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 17 June 2012.
</i>
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