cycle_brent, a Python code which analyzes a cycle in an iterated function sequence using Brent'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 Brent's method to determine the values of MU and LAM, given F and X0.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
cycle_brent is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
cycle_floyd, a Python code 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.