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Ball convergence for an eighth order efficient method under weak conditions in Banach spaces
(Springer Nature, 2017) Argyros, I.K.; George, S.; Erappa, S.M.
We present a local convergence analysis of an eighth order- iterative method in order to approximate a locally unique solution of an equation in Banach space setting. Earlier studies have used hypotheses up to the fourth 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. © 2016, Sociedad Española de Matemática Aplicada.
On the convergence of the sixth order Homeier like method in Banach spaces
(Erdal Karapinar, 2022) Suma, P.B.; Erappa, S.M.; George, S.
A sixth order Homeier-like method is introduced for approximating a solution of the non-linear equation in Banach space. Assumptions only on first and second derivatives are used to obtain a sixth order convergence. Our proof does not depend on Taylor series expansions as in the earlier studies for the similar methods. © 2022, Erdal Karapinar. All rights reserved.