Local convergence of a Newton–Traub composition in Banach spaces

Abstract

We study the local convergence of a sixth-order Newton–Traub method to approximate a locally-unique solution of a system of nonlinear equations. Earlier studies show convergence under hypotheses on the sixth derivative or even higher, although only the first derivatives are used in the method. The convergence in this study is shown under hypotheses only on the first derivative. Our analysis avoids the usual Taylor expansions requiring higher order derivatives but uses generalized Lipschitz–Holder-type conditions only on the first derivative. Moreover, our new approach provides computable radius of convergence as well as error bounds on the distances involved and estimates on the uniqueness of the solution based on some functions appearing in these generalized conditions. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives which may not exist or may be very expensive or impossible to compute. Therefore, we do not know how close to the solution the initial point should be for convergence of the method. That is the initial point is a shot in the dark with the approaches using Taylor expansions. The convergence order is computed using computational order of convergence or approximate computational order of convergence which do not require usage of higher derivatives. This technique can be applied to any iterative method using Taylor expansions involving high order derivatives. In this sense the applicability of the method is expanded. Finally, numerical examples are provided to show that the present results apply to solve equations in cases where earlier results cannot apply.