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<html>
<head>
<title>
CYCLE_BRENT - Cycle Detection by Brent's Method
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
CYCLE_BRENT <br> Cycle Detection by Brent's Method
</h1>
<hr>
<p>
<b>CYCLE_BRENT</b>
is a MATLAB library which
analyzes a cycle in an iterated function sequence using Brent's method.
</p>
<p>
Suppose we a repeatedly apply a function f(), starting with the argument
x0, then f(x0), f(f(x0)) and so on. Suppose that the range of f is finite.
Then eventually the iteration must reach a cycle. Once the cycle is reached,
succeeding values stay within that cycle.
</p>
<p>
Starting at x0, there is a "nearest element" of the cycle, which is
reached after MU applications of f.
</p>
<p>
Once the cycle is entered, the cycle has a length LAM, which is the number
of steps required to first return to a given value.
</p>
<p>
This function uses Brent's method to determine the values of MU and LAM,
given F and X0.
</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>CYCLE_BRENT</b> is available in
<a href = "../../c_src/cycle_brent/cycle_brent.html">a C version</a> and
<a href = "../../cpp_src/cycle_brent/cycle_brent.html">a C++ version</a> and
<a href = "../../f77_src/cycle_brent/cycle_brent.html">a FORTRAN77 version</a> and
<a href = "../../f_src/cycle_brent/cycle_brent.html">a FORTRAN90 version</a> and
<a href = "../../m_src/cycle_brent/cycle_brent.html">a MATLAB version</a> and
<a href = "../../py_src/cycle_brent/cycle_brent.html">a PYTHON version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/cycle_floyd/cycle_floyd.html">
CYCLE_FLOYD</a>,
a MATLAB library which
carries out an iterated function evaluation, and seeks to determine the
nearest element of a cycle, and the cycle's length, using Floyd's method.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Richard Brent,<br>
An improved Monte Carlo factorization algorithm,<br>
BIT,<br>
Volume 20, Number 2, 1980, pages 176-184.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "cycle_brent.m">cycle_brent.m</a>,
finds a cycle in an iterated mapping using Brent's method.
</li>
<li>
<a href = "timestamp.m">timestamp.m</a>,
prints the YMDHMS date as a timestamp.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "cycle_brent_test.m">cycle_brent_test.m</a>,
a sample calling program.
</li>
<li>
<a href = "cycle_brent_test_output.txt">cycle_brent_test_output.txt</a>,
the output file.
</li>
<li>
<a href = "cycle_brent_test01.m">cycle_brent_test01.m</a>,
tests cycle_brent on f1.
</li>
<li>
<a href = "f1.m">f1.m</a>,
test function 1.
</li>
<li>
<a href = "cycle_brent_test02.m">cycle_brent_test02.m</a>,
tests cycle_brent on f2.
</li>
<li>
<a href = "f2.m">f2.m</a>,
test function 2.
</li>
<li>
<a href = "cycle_brent_test03.m">cycle_brent_test03.m</a>,
tests cycle_brent on f3.
</li>
<li>
<a href = "f3.m">f3.m</a>,
test function 3.
</li>
<li>
<a href = "cycle_brent_test04.m">cycle_brent_test04.m</a>,
tests cycle_brent on f4.
</li>
<li>
<a href = "f4.m">f4.m</a>,
test function 4.
</li>
<li>
<a href = "cycle_brent_test05.m">cycle_brent_test05.m</a>,
tests cycle_brent on f5.
</li>
<li>
<a href = "f5.m">f5.m</a>,
test function 5.
</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 17 June 2012.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>