Developments on the Convergence Analysis of Newton-Kantorovich Method for Solving Nonlinear Equations

Abstract

Developments are presented for the semi-local convergence of Newton’s method to solve Banach space-valued nonlinear equations. By utilizing a new methodology, we provide a finer convergence analysis with no additional conditions than in earlier results. In particular, this is done by introducing the center-Lipschitz condition by which we construct a stricter domain than the original domain of the operator. Then, the Lipschitz constants in the new domain are at least as small as the original constants leading to weaker sufficient convergence criteria, tighter error bounds on the error distances involved, and a piece of better information on the location of the solution. These benefits are obtained under the same computational cost since in practice the computation of the original constants requires the computation of the new constants as special cases. The same benefits are obtained if the Lipschitz conditions are replaced by Hölder conditions or even more general ω− continuity conditions. This methodology can be applied to other methods using such as the Secant, Stirling’s Newton-like, and other methods along the same lines. Numerical examples indicate that the new results can be utilized to solve nonlinear equations, but not earlier ones.