PHoM (Polyhedral Homotopy Continuation Method for Polynomial Systems)

Important notice: Currently, we do not have any plan to release a new version of PHoM in the near future, and we will no longer provide service for PHoM package. This webpage will not be available from Dec. 31, 2008. For solving polynomial systems by the polyhedral homotopy method, we recommend HOME4PS-2.0.

PHoM is a software package for a polyhedral homotopy continuation method of finding all isolated solutions of a system of n polynomial equations f(x) = 0 in n-dimensional complex vector variable x. It is implemented in C++ language using the LAPACK for matrix computation. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f(x) = 0. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of f(x) = 0. The third module Verify checks whether all isolated solutions of f(x)=0 have been approximated correctly. For more details, see the paper below, which also serves as a user's guide of PHoM.

Takayuki Gunji, Sunyoung Kim, Masakazu Kojima, Akiko Takeda, Katsuki Fujisawa and Tomohiko Mizutani, PHoM -- a Polyhedral Homotopy Continuation Method for Polynomial Systems,'' Research Report B-386, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Meguro, Tokyo 152-8552, Japan, December 2002, revised January 2003. ps file, ps.gz file, pdf file.

The C++ program source codes and numerical results below are available.

Each software of PHoM is distributed under the GNU GPL (General Public License).

Some Other Results on Polynomial Systems

• A. Takeda, M. Kojima and K. Fujisawa ``Enumeration of All Solutions of a Combinatorial Linear Inequality System Arising from the Polyhedral Homotopy Continuation Method'', Journal of the Operations Research Society of Japan Vol.45, No.1, 64-82 (2002).
• Y. Dai, S. Kim and M. Kojima, ``Computing All Nonsingular Solutions of Cyclic-n Polynomial Using Polyhedral Homotopy Continuation Methods'', September 2001. Revised May 2002. To appear in Journal of Computational and Applied Mathematics. ps file, ps.gz file, pdf file.
• Solution Information on Some Polynomial Systems (Cyclic-n Polynomial, Economic Polynomial)
• S. Kim and M. Kojima ``CMPSm : Continuation Methods for Polynomial Systems (MATLAB Version)''