PHoM – a Polyhedral Homotopy Continuation Method for Polynomial Systems

  • Authors:

  • Takayuki Gunji;Sunyoung Kim;Masakazu Kojima;Akiko Takeda;Katsuki Fujisawa;Tomohiko Mizutani

  • Affiliations:

  • Tokyo Institute of Technology, Department of Mathematical and Computing Sciences, 2-12-1 Oh-Okayama, 152-8552, Meguro-ku Tokyo, Japan;Ewha Women’s University, Department of Mathematics, 11-1 Dahyun-dong, 120-750, Seoul, Sudaemoon-gu, Korea;Tokyo Institute of Technology, Department of Mathematical and Computing Sciences, 2-12-1 Oh-Okayama, 152-8552, Meguro-ku Tokyo, Japan;Tokyo Institute of Technology, Department of Mathematical and Computing Sciences, 2-12-1 Oh-Okayama, 152-8552, Meguro-ku Tokyo, Japan;Denki University, Department of Mathematical Sciences Tokyo, Saiwai-ku, 350-0394, Hatoyama Saitama, Sudaemoon-gu, Japan;1, Komukai Toshiba-cho, Toshiba Corporation Corporate Research & Development Center, Saiwai-ku, 212-8582, Kawasaki, Sudaemoon-gu, Japan

  • Venue:
  • Computing
  • Year:
  • 2004

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Abstract

PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. 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. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.