The reliability of polyhedral homotopy continuation methods for solving a polynomial system becomes increasingly important as the dimension of the polynomial system increases. High powers of the homotopy continuation parameter $t$ and ill-conditioned Jacobian matrices encountered in tracing of homotopy paths affect the numerical stability. We present modified homotopy functions with a new homotopy continuation parameter $s$ and various scaling strategies to enhance the numerical stability. Advantages of employing the new homotopy parameter $s$ are discussed. Numerical results are included to illustrate improved performance of the presented techniques.
Research report B-380, Department of Mathematical and Computing Sciences, Tokyo Institute of Technoloy 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8552 Japan March/2003