Faculty Publications
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Item Ball convergence theorems for unified three step Newton-like methods of high convergence order(Touch Briefings jonathan.mckenna@touchbriefings.com, 2015) Argyros, I.K.; George, S.We present a local convergence analysis for eighth-order variants of Newton's method in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as [7]-[11], [20] using hypotheses up to the seventh derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study. © CSP - Cambridge, UK; I&S - Florida, USA, 2015.Item Ball convergence for a Newton-steffensen-type third-order method(International Publications internationalpubls@yahoo.com, 2015) Argyros, I.K.; George, S.We present a local convergence analysis for a composite Newton-Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as [1], [5]-[28] using hypotheses up to the second derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.Item Local convergence for some third-order iterative methods under weak conditions(Korean Mathematical Society kms@kms.or.kr, 2016) Argyros, I.K.; Cho, Y.J.; George, S.The solutions of equations are usually found using iterative methods whose convergence order is determined by Taylor expansions. In particular, the local convergence of the method we study in this paper is shown under hypotheses reaching the third derivative of the operator involved. These hypotheses limit the applicability of the method. In our study we show convergence of the method using only the first derivative. This way we expand the applicability of the method. Numerical examples show the applicability of our results in cases earlier results cannot. © 2016 Korean Mathematical Society.Item On a result by Dennis and Schnabel for Newton's method: Further improvements(Elsevier Ltd, 2016) Argyros, I.K.; George, S.We improve local convergence results for Newton's method by defining a more precise domain where the Newton iterates lie than in earlier studies using Dennis and Schnabel-type techniques. A numerical example is presented to show that the new convergence radii are larger and new error bounds are more precise than the earlier ones. © 2015 Elsevier Ltd. All rights reserved.Item Ball convergence theorems for Maheshwari-type eighth-order methods under weak conditions(Springer International Publishing, 2016) Argyros, I.K.; George, S.We present a local convergence analysis for a family of Maheshwari-type eighth-order methods in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as Cordero et al. (J Comput Appl Math 291(1):348–357, 2016), Maheshwari (Appl Math Comput 211:283–391, 2009), Petkovic et al. (Multipoint methods for solving nonlinear equations. Elsevier, Amsterdam, 2013) using hypotheses up to the seventh derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study. © 2015, Instituto de Matemática e Estatística da Universidade de São Paulo.Item Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions(Springer-Verlag Italia s.r.l., 2016) Argyros, I.K.; George, S.We present a local convergence analysis of a sixth order iterative method for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Sharma et al. (Appl Math Comput 190:111–115, 2007) have shown convergence of these methods under hypotheses up to the fifth derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study. © 2015, Springer-Verlag Italia.Item Ball convergence theorem for a Steffensen-type third-order method(Universidad Nacional de Colombia revcolamt@scm.org.co, 2017) Argyros, I.K.; George, S.We present a local convergence analysis for a family of Steffensen- type third-order methods in order to approximate a solution of a nonlinear equation. We use hypothesis up to the first derivative in contrast to earlier studies such as [2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] using hypotheses up to the fourth derivative. This way the applicability of these methods is extended under weaker hypothesis. More- over the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.Item Ball convergence of some iterative methods for nonlinear equations in Banach space under weak conditions(Springer-Verlag Italia s.r.l., 2018) Argyros, I.K.; George, S.The aim of this paper is to expand the applicability of a fast iterative method in a Banach space setting. Moreover, we provide computable radius of convergence, error bounds on the distances involved and a uniqueness of the solution result based on Lipschitz-type functions not given before. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. The convegence order is determined using the computational order of convergence or the approximate order of convergence. Numerical examples where earlier results cannot be applied to solve equations but our results can be applied are also given in this study. © 2017, Springer-Verlag Italia S.r.l.Item Ball convergence theorems for J. Chen’s one step third-order iterative methods under weak conditions(International Publications internationalpubls@yahoo.com, 2020) Argyros, I.K.; George, S.We present a local convergence analysis for J. Chen’s one step third-order iterative method in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as [8] using hypotheses up to the third derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples where earlier results cannot be used to solve equations but our results can be used are also presented in this study. © 2020, International Publications. All rights reserved.Item Ball convergence theorems for iterative methods under weak conditions(International Publications internationalpubls@yahoo.com, 2020) George, S.; Argyros, I.K.We revisit a Weerakoon type iterative method for solving equations. Earlier studies have used higher order derivatives not appearing in the method for the convergence analysis. But this way the usage of the method is restricted though it may converge. That is why in order to extend its applicability, we only use hypotheses on the first derivative that actually is on the method. The fifth order of convergence has also been carried out on the finite dimensional Euclidean space. But our analysis involves more general setting of Banach space valued operators. Our idea can be used to extend the applicability of other methods along the same lines. © 2020, International Publications. All rights reserved.