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<html>
<head>
<title>
ASA006 - Cholesky Factor of a Positive Definite Symmetric Matrix
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
ASA006 <br> Cholesky Factor of a Positive Definite Symmetric Matrix
</h1>
<hr>
<p>
<b>ASA006</b>
is a MATLAB program which
computes the Cholesky factor of a positive definite symmetric matrix.
</p>
<p>
<b>ASA006</b> is Applied Statistics Algorithm 6. Source code for many
Applied Statistics Algorithms is available through
<a href = "http://lib.stat.cmu.edu/apstat">STATLIB</a>.
</p>
<p>
If <b>A</b> is a positive definite symmetric matrix, then there is an
upper triangular matrix <b>U</b> with the property that
<blockquote><b>
A = U' * U
</b></blockquote>
The matrix <b>U</b> is known as the <i>Cholesky factor</i> of <b>A</b>,
and can be used to easily solve linear systems involving <b>A</b>
or compute the inverse of <b>A</b>.
</p>
<p>
The algorithm implemented here uses a compressed storage for both the
matrix <b>A</b> and the factor <b>U</b>. This saves some storage, but
can make computations a little awkward.
</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>ASA006</b> is available in
<a href = "../../c_src/asa006/asa006.html">a C version</a> and
<a href = "../../cpp_src/asa006/asa006.html">a C++ version</a> and
<a href = "../../f77_src/asa006/asa006.html">a FORTRAN77 version</a> and
<a href = "../../f_src/asa006/asa006.html">a FORTRAN90 version</a> and
<a href = "../../m_src/asa006/asa006.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/asa007/asa007.html">
ASA007</a>,
a MATLAB library which
computes the inverse of a symmetric positive definite matrix,
and uses a version of ASA006 for for Cholesky factorization.
</p>
<p>
<a href = "../../m_src/asa047/asa047.html">
ASA047</a>,
a MATLAB library which
implements the Nelder-Mead minimization algorithm,
and uses a version of ASA006 for Cholesky factorization.
</p>
<p>
<a href = "../../f_src/lapack/lapack.html">
LAPACK</a>,
a FORTRAN90 library which
includes routines for Cholesky factorization.
</p>
<p>
<a href = "../../m_src/linpack/linpack.html">
LINPACK</a>,
a MATLAB library which
includes routines for Cholesky factorization.
</p>
<p>
<a href = "../../f77_src/nl2sol/nl2sol.html">
NL2SOL</a>,
a FORTRAN77 library which
solves nonlinear least squares problems, and
includes routines for Cholesky factorization.
</p>
<p>
<a href = "../../f77_src/pppack/pppack.html">
PPPACK</a>,
a FORTRAN77 library which
carries out piecewise polynomial interpolation and
includes routines for Cholesky factorization.
</p>
<p>
<a href = "../../m_src/toeplitz_cholesky/toeplitz_cholesky.html">
TOEPLITZ_CHOLESKY</a>,
a MATLAB library which
computes the Cholesky factorization of a nonnegative definite symmetric
Toeplitz matrix.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
PR Freeman,<br>
Remark AS R44:
A Remark on AS 6 and AS7: Triangular decomposition of a symmetric matrix
and Inversion of a positive semi-definite symmetric matrix,<br>
Applied Statistics,<br>
Volume 31, Number 3, 1982, pages 336-339.
</li>
<li>
Michael Healy,<br>
Algorithm AS 6:
Triangular decomposition of a symmetric matrix,<br>
Applied Statistics,<br>
Volume 17, Number 2, 1968, pages 195-197.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cholesky.m">cholesky.m</a>
computes the Cholesky factorization of a PDS matrix.
</li>
<li>
<a href = "subchl.m">subchl.m</a>
computes the Cholesky factorization of a (subset of a) PDS matrix.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>
prints out the current YMDHMS date as a timestamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "asa006_test.m">asa006_test.m</a>,
a sample calling program.
</li>
<li>
<a href = "asa006_test01.m">asa006_test01.m</a>,
tests CHOLESKY on a simple matrix.
</li>
<li>
<a href = "asa006_test02.m">asa006_test02.m</a>,
tests CHOLESKY on the Hilbert matrix.
</li>
<li>
<a href = "asa006_test03.m">asa006_test03.m</a>,
tests SUBCHL on the Hilbert matrix.
</li>
<li>
<a href = "asa006_test_output.txt">asa006_test_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB source codes</a>.
</p>
<hr>
<i>
Last revised on 26 February 2010.
</i>
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