CYCLE_BRENT
Cycle Detection by Brent's Method

CYCLE_BRENT is a FORTRAN77 library 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.

Licensing:

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

Languages:

CYCLE_BRENT is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version and a PYTHON version.

Related Data and Programs:

CYCLE_FLOYD, a FORTRAN77 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.

Reference:

1. Richard Brent,
   An improved Monte Carlo factorization algorithm,
   BIT,
   Volume 20, Number 2, 1980, pages 176-184.

Source Code:

• cycle_brent.f, the source code.
• cycle_brent.sh, BASH commands to compile the source code.

Examples and Tests:

• cycle_brent_prb.f, a sample calling program.
• cycle_brent_prb.sh, BASH commands to compile and run the sample program.
• cycle_brent_prb_output.txt, the output file.

List of Routines:

• CYCLE_BRENT finds a cycle in an iterated mapping using Brent's method.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN77 source codes.