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<html>
<head>
<title>
SVD_TRUNCATED - The Truncated Singular Value Decomposition
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
SVD_TRUNCATED <br> The Truncated Singular Value Decomposition
</h1>
<hr>
<p>
<b>SVD_TRUNCATED</b>
is a FORTRAN90 program which
demonstrates the computation of the reduced or truncated
Singular Value Decomposition (SVD)
of an M by N rectangular matrix, in cases where M < N or N < M.
</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>
<p>
It is often the case that the matrix A has one dimension much bigger than the other.
For instance, M = 3 and N = 10,000 might be such a case.
For such examples, much of the computation and memory required for the standard SVD
may not actually be needed. Instead, a truncated, or reduced version is appropriate.
It will be computed faster, and require less memory to store the data.
</p>
<p>
If M < N, we have the "truncated V" SVD:
<pre>
A(mxn) = U(mxm) * Sm(mxm) * Vm'(nxm)
</pre>
Notice that, for our example, we will have to compute and store a Vm of size
30,000 instead of a V of size 1,000,000 entries.
</p>
<p>
If N < M, we have the "truncated U" SVD:
<pre>
A(mxn) = Un(mxn) * Sn(nxn) * V'(nxn)
</pre>
Similarly, in this case, the computation and storage of Un can be much reduced
from that of U.
</p>
<p>
The LAPACK routines CGESVD, DGESVD, SGESVD and ZGESVD compute the SVD for
a rectangular matrix in single or double precision, real or complex arithmetic.
These routines include some options that allow you to request a reduced or
truncated SVD computation, but the exact details of how to do this may be
a little obscure. This program demonstrates how it is done.
</p>
<p>
In order to compile and load this program, you need to have access to a
copy of LAPACK.
</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_TRUNCATED</b> is available in
<a href = "../../cpp_src/svd_truncated/svd_truncated.html">a C++ version</a> and
<a href = "../../f_src/svd_truncated/svd_truncated.html">a FORTRAN90 version</a> and
<a href = "../../m_src/svd_truncated/svd_truncated.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 = "../../f_src/linpack_d/linpack_d.html">
LINPACK_D</a>,
a FORTRAN90 library which
solves linear systems using double precision real arithmetic;
</p>
<p>
<a href = "../../f_src/svd_basis/svd_basis.html">
SVD_BASIS</a>,
a FORTRAN90 program which
computes a reduced basis for a collection of data vectors using the SVD.
</p>
<p>
<a href = "../../f_src/svd_demo/svd_demo.html">
SVD_DEMO</a>,
a FORTRAN90 program which
demonstrates the singular value decomposition (SVD) for a simple example.
</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>
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_truncated.f90">svd_truncated.f90</a>, the source code.
</li>
<li>
<a href = "svd_truncated.sh">svd_truncated.sh</a>,
commands to compile and load the source code.
</li>
<li>
<a href = "svd_truncated_output.txt">svd_truncated_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program for SVD_TRUNCATED.
</li>
<li>
<b>SVD_TRUNCATED_U_TEST</b> tests SVD_TRUNCATED_U.
</li>
<li>
<b>SVD_TRUNCATED_U</b> computes the SVD when N <= M.
</li>
<li>
<b>SVD_TRUNCATED_V_TEST</b> tests SVD_TRUNCATED_V.
</li>
<li>
<b>SVD_TRUNCATED_V</b> computes the SVD when M <= N.
</li>
<li>
<b>R8MAT_PRINT</b> prints an R8MAT.
</li>
<li>
<b>R8MAT_PRINT_SOME</b> prints some of an R8MAT.
</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 = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 19 March 2012.
</i>
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