For the HQRB method, the main program is lyapunov.m, which is written as a MATLAB function file. The function lyapunov computes the trajectory and the Lyapunov characteristic exponents for discrete systems using the Householder QR Based Method.  For more information, see "An Efficient QR Based Method for the Computation of Lyapunov Exponents" , by Hubertus F. von Bremen, Firdaus E. Udwadia, and Wlodek Proskurowski, Physica D, vol. 101,  pp. 1-16, 1997.

For the HQRBp1 and HQRBp2 methods the main functions are thqrbp1.m and thqrbp2 respectively.  For more information on the methods, see "A note on the computation of the largest p LCEs of discrete dynamical systems", by Firdaus E. Udwadia, Hubertus F. von Bremen and Wlodek Proskurowski, Applied Mathematics and Computation, vol. 114, Nos. 2-3, September 2000 (to appear).

In order to run the function lyapunov.m (or thqrbp1.m or thqrbp2.m) you need two additional user defined function files.  One of the files has to specify the map defining the trajectory, and the other the tangent map to the trajectory. Sample files defining the trajectory and the tangent map for a coupled logistics map are provided in the files map.m and tanmap.m respectively.  To become familiar with the program, run it using the sample files map.m and tanmap.m (these files can be adapted to other dynamical systems).

If the dimension of a system is large, it may take a long time to compute the LCE's.  In order to get an estimate of the running time, for the first run of a "new" dynamical system, you may want to set the parameters Nskip and Ntot to relatively small values.

The functions lyapunov.m, thqrbp1.m and thqrbp2.m include the information needed to run them.  Each one of the main programs (Lyapunov.m, thqrbp1.m and thqrbp2.m) has a help portion available (accessible through typing:  help filename from the MATLAB prompt). For example, on the MATLAB command window you can type: help lyapunov, and a help file will appear (if it does not, make sure that the file lyapunov.m is in your MATLAB path).