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Author: search.filters.author.Padikkal, P.Subject: search.filters.subject.Local convergence

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    On the local convergence of newton-like methods with fourth and fifth order of convergence under hypotheses only on the first fréchet derivative
    (Institute of Mathematics nsjom@dmi.uns.ac.rs, 2017) Argyros, I.K.; Padikkal, P.; George, S.
    We present a local convergence analysis of several Newton-like methods with fourth and fifth order of convergence in order to approximate a locally unique solution of an equation in Banach space setting. Earlier studies have used hypotheses up to the fifth derivative although only the first derivative appears in the definition of these methods. In this study we only use the hypothesis on the first derivative. This way we expand the applicability of these methods. Moreover, we provide a radius of convergence, a uniqueness ball and computable error bounds based on Lipschitz constants. Numerical examples computing the radii of the convergence balls as well as examples where earlier results cannot apply to solve equations but our results can apply are also given in this study. © 2017, Institute of Mathematics. All rights reserved.
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    Ball Convergence for Second Derivative Free Methods in Banach Space
    (Springer, 2017) Argyros, I.K.; Padikkal, P.; George, S.
    Hueso et al. (Appl Math Comput 211:190–197, 2009) considered a third and fourth order iterative methods for nonlinear systems. The methods were shown to of order third and fourth if the operator equation is defined on the j-dimensional Euclidean space (Hueso et al. in Appl Math Comput 211:190–197, 2009). The order of convergence was shown using hypotheses up to the third Fréchet derivative of the operator involved although only the first derivative appears in these methods. In the present study we only use hypotheses on the first Fréchet-derivative. This way the applicability of these methods is expanded. Moreover we present a radius of convergence a uniqueness result and computable error bounds based on Lipschitz constants. Numerical examples are also presented in this study. © 2015, Springer India Pvt. Ltd.