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<html>
<head>
<title>
TOEPLITZ_CHOLESKY - Cholesky Factor of Nonnegative Definite Toeplitz Matrix
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
TOEPLITZ_CHOLESKY <br> Cholesky Factor of Nonnegative Definite Toeplitz Matrix
</h1>
<hr>
<p>
<b>TOEPLITZ_CHOLESKY</b>
is a FORTRAN90 library which
computes the Cholesky factorization of a nonnegative definite symmetric
Toeplitz matrix.
</p>
<p>
A Toeplitz matrix is a matrix which is constant along all diagonals. A schematic
of a 3x4 Toeplitz matrix would be
<pre>
a b c d
e a b c
f e a b
</pre>
</p>
<p>
A symmetric matrix is a matrix with N rows and N columns, such that
A(I,J) = A(J,I) for all indices I and J. All the eigenvalues of a symmetric
matrix are real.
</p>
<p>
A symmetric Toeplitz matrix is a matrix which is symmetric and Toeplitz. A schematic
of a 4x4 symmetric Toeplitz matrix would be
<pre>
a b c d
b a b c
c b a b
d c b a
</pre>
</p>
<p>
A nonnegative definite symmetric matrix A is a symmetric matrix whose eigenvalues
are all nonnegative.
</p>
<p>
Given a nonnegative definite symmetric matrix A, the upper Cholesky factor R
is an upper triangular matrix such that A = R' * R; the lower Cholesky factor
L is a lower triangular matrix such that A = L L'. Obviously, L = R'.
</p>
<p>
A Toeplitz matrix can be represented in a compressed format that stores
the first row and the first column (omitting the first entry). One
convenient format would be to create the 2xN array G as follows:
<pre>
G(1,1:N) = A(1,1:N)
G(2,1) = 0.0
G(2,2:N) = A(2:N,1)
</pre>
</p>
<p>
A symmetric Toeplitz matrix can be represented in a compressed format that stores
just the first row.
</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>TOEPLITZ_CHOLESKY</b> is available in
<a href = "../../c_src/toeplitz_cholesky/toeplitz_cholesky.html">a C version</a> and
<a href = "../../cpp_src/toeplitz_cholesky/toeplitz_cholesky.html">a C++ version</a> and
<a href = "../../f77_src/toeplitz_cholesky/toeplitz_cholesky.html">a FORTRAN77 version</a> and
<a href = "../../f_src/toeplitz_cholesky/toeplitz_cholesky.html">a FORTRAN90 version</a> and
<a href = "../../m_src/toeplitz_cholesky/toeplitz_cholesky.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/asa006/asa006.html">
ASA006</a>,
a FORTRAN90 library which
computes the Cholesky factorization of a symmetric positive definite
matrix, by Michael Healy.
This is a MATLAB version of Applied Statistics Algorithm 6;
</p>
<p>
<a href = "../../f_src/linpack_d/linpack_d.html">
LINPACK_D</a>,
a FORTRAN90 library which
factors and solves linear systems using double precision real arithmetic,
by Jack Dongarra, Jim Bunch, Cleve Moler, Pete Stewart.
</p>
<p>
<a href = "../../f_src/linplus/linplus.html">
LINPLUS</a>,
a FORTRAN90 library which
carries out simple manipulations of matrices in a variety of formats,
including matrices stored in banded, border-banded, circulant,
lower triangular, pentadiagonal, sparse, symmetric,
toeplitz, tridiagonal, upper triangular and vandermonde formats.
</p>
<p>
<a href = "../../f_src/toeplitz/toeplitz.html">
TOEPLITZ</a>,
a FORTRAN90 library
solves a variety of Toeplitz and circulant linear systems.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Michael Stewart,<br>
Cholesky factorization of semi-definite Toeplitz matrices.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "toeplitz_cholesky.f90">toeplitz_cholesky.f90</a>, the source code.
</li>
<li>
<a href = "toeplitz_cholesky.sh">toeplitz_cholesky.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "toeplitz_cholesky_prb.f90">toeplitz_cholesky_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "toeplitz_cholesky_prb.sh">toeplitz_cholesky_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "toeplitz_cholesky_prb_output.txt">toeplitz_cholesky_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<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>
<li>
<b>TOEP_CHOLESKY_LOWER:</b> lower Cholesky factor of a Toeplitz matrix.
</li>
<li>
<b>TOEP_CHOLESKY_UPPER:</b> upper Cholesky factor of a Toeplitz matrix.
</li>
<li>
<b>TOEPLITZ_CHOLESKY_LOWER:</b> lower Cholesky factor of a Toeplitz matrix.
</li>
<li>
<b>TOEPLITZ_CHOLESKY_UPPER:</b> upper Cholesky factor of a Toeplitz matrix.
</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 modified on 13 November 2012.
</i>
<!-- John Burkardt -->
</body>
</html>