Convergent regions of Newton homotopy methods for nonlinear systems: theory and computational applications

Abstract

This paper introduces the concept of the convergent region of the Newton homotopy method. It is shown that convergent regions of the Newton homotopy method are equal to the stability regions for the Newton flow x/spl dot/=-adj(DF(x))F(x). A quite complete algebraic characterization of a convergent region and its boundary for a large class of nonlinear systems is derived and this characterization, which is explicit and computationally feasible, leads to the development of a numerical method to determine the convergent region and to the construction of simple criteria to avoid revisitations of the same solutions from different initial guesses. Two examples are given to illustrate the theoretical prediction.