CYCLE_FLOYD
Cycle Detection by Floyd's Method

CYCLE_FLOYD is a Python library which analyzes a cycle in an iterated function sequence using Floyd's method.

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.

Starting at x0, there is a "nearest element" of the cycle, which is reached after MU applications of f.

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.

This function uses Floyd's method to determine the values of MU and LAM, given F and X0.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

CYCLE_FLOYD is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.

Related Data and Programs:

CYCLE_BRENT, a Python library which carries out an iterated function evaluation, and seeks to determine the nearest element of a cycle, and the cycle's length, using Brent's method.

Reference:

1. Donald Knuth,
   The Art of Computer Programming,
   Volume 2, Seminumerical Algorithms,
   Third Edition,
   Addison Wesley, 1997,
   ISBN: 0201896842,
   LC: QA76.6.K64.

Source Code:

• cycle_floyd.py, finds a cycle in an iterated mapping using Floyd's method.
• cycle_floyd_test.sh, a BASH script that runs the sample calling program.
• cycle_floyd_test.txt, the output file.

You can go up one level to the Python source codes.