Numerical Stablity of Path Tracing in Polyhedral Homotopy Continuation Methods

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 the improved performance of the presented techniques. AMS subject classification. Primary: 65H10 Systems of equations Secondary: 65H20 Global methods, including homotopy approaches