Faculty Publications
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Item Local convergence for multi-point-parametric Chebyshev-Halley-type methods of high convergence order(Elsevier, 2015) Argyros, I.K.; George, S.; Magreñán Ruiz, Á.A.We present a local convergence analysis for general multi-point-Chebyshev-Halley-type methods (MMCHTM) of high convergence order in order to approximate a solution of an equation in a Banach space setting. MMCHTM includes earlier methods given by others as special cases. The convergence ball for a class of MMCHTM methods is obtained under weaker hypotheses than before. Numerical examples are also presented in this study. © 2014 Elsevier B.V. All rights reserved.Item Ball convergence comparison between three iterative methods in Banach space under hypothese only on the first derivative(Elsevier Inc. usjcs@elsevier.com, 2015) Argyros, I.K.; George, S.Abstract We present a convergence ball comparison between three iterative methods for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The convergence ball and error estimates are given for these methods under hypotheses only on the first Fréchet derivative in contrast to earlier studies such as Adomian (1994) [1], Babajee et al. (2008) [13], Cordero and Torregrosa (2007) [17], Cordero et al. [18], Darvishi and Barati (2007) [19], using hypotheses reaching up to the fourth Fréchet derivative although only the first derivative appears in these methods. This way we expand the applicability of these methods. Numerical examples are also presented in this study. © 2015 Elsevier Inc.Item Unified convergence domains of Newton-like methods for solving operator equations(Elsevier Inc. usjcs@elsevier.com, 2016) Argyros, I.K.; George, S.We present a unified semilocal convergence analysis in order to approximate a locally unique zero of an operator equation in a Banach space setting. Using our new idea of restricted convergence domains we generate smaller Lipschitz constants than in earlier studies leading to the following advantages: weaker sufficient convergence criteria, tighter error estimates on the distances involved and an at least as precise information on the location of the zero. Hence, the applicability of these methods is extended. These advantages are obtained under the same cost on the parameters involved. Numerical examples where the old sufficient convergence criteria cannot apply to solve equations but the new criteria can apply are also provided in this study. © 2016 Elsevier Inc. All rights reserved.Item On the convergence of Broyden's method with regularity continuous divided differences and restricted convergence domains(Mathematical Research Press, 2017) Argyros, I.K.; George, S.We present a semilocal convergence analysis for Broyden's method with regularly continuous divided differences in a Banach/Hilbert space setting. By using: center-Lipschitz-type conditions defining restricted convergence domains, at least as weak hypotheses and the same computational cost as in [6] we provide a new convergence analysis for Broyden's method with the following advantages: larger convergence domain; finer error bounds on the distances involved, and at least as precise information on the location of the solution. © 2017 Journal of Nonlinear Functional Analysis.Item Local convergence of a fast Steffensen-type method on Banach space under weak conditions(Inderscience Publishers, 2017) Argyros, I.K.; George, S.This paper is devoted to the study of the seventh-order Steffensen-type methods for solving nonlinear equations in Banach spaces. Using the idea of a restricted convergence domain, we extended the applicability of the seventh-order Steffensen-type methods. Our convergence conditions are weaker than the conditions used in the earlier studies. Numerical examples are also given in this study. © © 2017 Inderscience Enterprises Ltd.Item Local Convergence of Jarratt-Type Methods with Less Computation of Inversion Under Weak Conditions(Taylor and Francis Ltd., 2017) Argyros, I.K.; George, S.We present a local convergence analysis for Jarratt-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Earlier studies cannot be used to solve equations using such methods. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study. © 2017, © Vilnius Gediminas Technical University, 2017.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 Local convergence for an eighth order method for solving equations and systems of equations(De Gruyter Open Ltd, 2019) Argyros, I.K.; George, S.The aim of this study is to extend the applicability of an eighth convergence order method from the k-dimensional Euclidean space to a Banach space setting. We use hypotheses only on the first derivative to show the local convergence of the method. Earlier studies use hypotheses up to the eighth derivative although only the first derivative and a divided difference of order one appear in the method. Moreover, we provide computable error bounds based on Lipschitz-type functions. © 2019 I.K Argyros and S. George.Item Unified convergence analysis of frozen Newton-like methods under generalized conditions(Elsevier B.V., 2019) Argyros, I.K.; George, S.The objective in this article is to present a unified convergence analysis of frozen Newton-like methods under generalized Lipschitz-type conditions for Banach space valued operators. We also use our new idea of restricted convergence domains, where we find a more precise location, where the iterates lie leading to at least as tight majorizing functions. Consequently, the new convergence criteria are weaker than in earlier works resulting to the expansion of the applicability of these methods. The conditions do not necessarily imply the differentiability of the operator involved. This way our method is suitable for solving equations and systems of equations. Numerical examples complete the presentation of this article. © 2018 Elsevier B.V.Item Kantorovich-Like Convergence Theorems for Newton’s Method Using Restricted Convergence Domains(Taylor and Francis Inc. 325 Chestnut St, Suite 800 Philadelphia PA 19106, 2019) Argyros, I.K.; George, S.The convergence set for Newton’s method is small in general using Lipschitz-type conditions. A center-Lipschitz-type condition is used to determine a subset of the convergence set containing the Newton iterates. The rest of the Lipschitz parameters and functions are then defined based on this subset instead of the usual convergence set. This way the resulting parameters and functions are more accurate than in earlier works leading to weaker sufficient semi-local convergence criteria. The novelty of the paper lies in the observation that the new Lipschitz-type functions are special cases of the ones given in earlier works. Therefore, no additional computational effort is required to obtain the new results. The results are applied to solve Hammerstein nonlinear integral equations of Chandrasekhar type in cases not covered by earlier works. © 2018, © 2019 Taylor & Francis Group, LLC.
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