OPTIMAL HOMOTOPY METHODS FOR SOLVING NONLINEAR-SYSTEMS .1. NONSINGULAR HOMOTOPY-PATHS

This paper deals with the monotone homotopy methods for solving the sy stem of nonlinear equations. In Sect. 2 the homotopy with a distance-m onotone homotopy path is discussed and it is proved that a homotopy wi th a given structure has a distance monotone homotopy path under some regular conditions. Then two structure-variable homotopy algorithms ca lled local straighten-up method and global straighten-up method are de veloped to approximate the optimal homotopy, and an adaptive step-size control strategy for these algorithms is proposed respectively. A new method for analyzing the convergence is presented, which is based on the geometrical properties of the algorithm. Then the convergence of t he algorithm is proved under certain regular conditions. Finally two n umerical examples are given to illustrate the effectiveness of the abo ve two algorithms.