The Convergence Ball and Error Analysis of the Relaxed Secant Method

A relaxed secant method is proposed. Radius estimate of the convergence ball of the relaxed secant method is attained for the nonlinear equation systems with Lipschitz continuous divided differences of first order. The error estimate is also established with matched convergence order. From the radius and error estimate, the relation between the radius and the speed of convergence is discussed with parameter. At last, some numerical examples are given

On the convergence of inexact two-point Newton-like methods on Banach spaces

We present a unified convergence analysis of Inexact Newton like methods in order to approximate a locally unique solution of a nonlinear operator equation containing a nondifferentiable term in a Banach space setting. The convergence conditions are more general and the error analysis more precise than in earlier studies such as (Argyros, 2007: Catinas, 2005; Catinas, 1994; Chen and Yamamoto, 1989: Dennis, 1968: Hernandez and Romero, 2005; Potra and Ptak, 1984; Rheinboldt, 1977). Special cases of our results can be used to find zeros of derivatives. Numerical examples are also provided in this study. (C) 2015 Elsevier Inc. All rights reserved