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

This paper deals with the monotone homotopy methods for solving the system of nonlinear equations. In Sect. 2 the homotopy with a distance-monotone homotopy path is discussed and it is proved that a homotopy with a given structure has a distance monotone homotopy path under some regular conditions. Then two structure-variable homotopy algorithms called local straighten-up method and global straighten-up method are developed 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 the algorithm is proved under certain regular conditions. Finally two numerical examples are given to illustrate the effectiveness of the above two algorithms.