On Newton-Iterative Methods for the Solution of Systems of Nonlinear Equations

In this paper we consider the local rates of convergence of Newton-iterative methods for the solution of systems of nonlinear equations. We show that under certain conditions on the inner, linear iterative method, Newton-iterative methods can be made to converge quadratically in a certain sense by computing a sufficient number of inner iterates at each step. As examples of this phenomenon, we consider the Newton-SOR and Newton-Richardson methods, particularly as applied to semilinear partial differential equations. Numerical results are included to illustrate the theory.