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

This paper introduces the concept of the convergent region of a solution of a general nonlinear equation using the Newton homotopy method. The question of whether an initial guess converges to the solution of our interest using the Newton homotopy method is investigated. It is shown that convergent regions of the Newton homotopy method are equal to stability regions of a corresponding Newton dynamic system. A necessary and sufficient condition for the adjacency of two solutions using the Newton homotopy method is derived. An algebraic characterization of a convergent region and its boundary for a large class of nonlinear systems is derived. This characterization is explicit and computationally feasible. A numerical method to determine the convergent region and to establish simple criteria to avoid revisits of the same solutions from different initial guesses is developed. It is shown that for general nonlinear systems or gradient systems, it is computationally infeasible to construct a set of initial guesses which converge to the set of all type-one equilibrium points on the stability boundary of a stable equilibrium point x_s from a finite number of function values and derivatives near x_s using the Newton homotopy method. Several examples are applied to illustrate the theoretical developments