New Kantorovich-Type Conditions for Halley's Method

Abstract

Two new semilocal convergence results of Newton-Kantorovich type are presented for the Halley method, where the usual convergence conditions, which appears in the literature, are relaxed. In one of them, it is supposed that the second derivative F″ of a nonlinear operator F satisfies ‖F″(x₀)‖ ≤ α instead of ‖F″(x)‖ ≤ M, for all x in a subset of the domain of F, where α and M are positive real constants. In the other one fewer convergence conditions are required than all the existing ones until now. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)