Journal Articles
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Item Extending the applicability of the inexact Newton-HSS method for solving large systems of nonlinear equations(Springer, 2020) Argyros, I.K.; George, S.; Senapati, K.We revisit the study of the semi-local convergence of the inexact Newton-HSS method (INHSS) introduced by Amiri et al. (2018), for solving large systems of nonlinear equations. In particular, first we present the correct convergence criterion, since the one in the preceding reference is incorrect. Secondly, we present an even weaker convergence criterion using our idea of recurrent functions. Moreover, the bound functions are compared. Finally, numerical examples are provided to show that the earlier convergence criteria are not satisfied but the new ones are satisfied. Hence, the applicability of the INHSS method is extended and under the same information as in the earlier studies. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.Item Local Convergence of Inexact Newton-Like Method under Weak Lipschitz Conditions(Springer, 2020) Argyros, I.K.; Cho, Y.J.; George, S.; Xiao, Y.The paper develops the local convergence of Inexact Newton-Like Method (INLM) for approximating solutions of nonlinear equations in Banach space setting. We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis. The obtained results compare favorably with earlier ones such as [7, 13, 14, 18, 19]. A numerical example is also provided. © 2020, Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences.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 On an iterative method without inverses of derivatives for solving equations(Erdal Karapinar, 2020) Argyros, I.K.; George, S.We present the semi-local convergence analysis of a Potra-type method to solve equations involving Banach space valued operators. The analysis is based on our ideas of recurrent functions and restricted convergence region. The study is completed using numerical examples. © 2020, Erdal Karapinar. All rights reserved.Item Ball convergence for a sixth-order multi-point method in banach spaces under weak conditions(Institute of Mathematics. Polish Academy of Sciences publ@impan.gov.pl, 2020) Argyros, I.K.; George, S.The aim of this paper is to extend the applicability of some high order iterative methods without using hypotheses on derivatives not appearing in those methods. Numerical examples are given where earlier convergence conditions are not satisfied but the new ones are satisfied. © Instytut Matematyczny PAN, 2020Item Extensions of kantorovich-type theorems for Newton's method(Institute of Mathematics. Polish Academy of Sciences publ@impan.gov.pl, 2020) Argyros, I.K.; George, S.; Sahu, D.R.We extend the applicability of Newton's method, so we can approximate a locally unique solution of a nonlinear equation in a Banach space setting in cases not covered before. To achieve this, we find a more precise set containing the Newton iterates than in earlier works. © Instytut Matematyczny PAN, 2020Item High Convergence Order Q-Step Methods for Solving Equations and Systems of Equations(Universal Wiser Publisher, 2020) Argyros, I.K.; George, S.The local convergence analysis of iterative methods is important since it demonstrates the degree of difficulty for choosing initial points. In the present study, we introduce generalized multi-step high order methods for solving nonlinear equations. The local convergence analysis is given using hypotheses only on the first derivative which actually appears in the methods in contrast to earlier works using hypotheses on higher order derivatives. This way we extend the applicability of these methods. The analysis includes the computable radius of convergence as well as error bounds based on Lipschitz-type conditions not given in earlier studies. Numerical examples conclude this study. © 2020, Ioannis K. Argyros et al.Item COMPARISON BETWEEN SOME SIXTH CONVERGENCE ORDER SOLVERS UNDER THE SAME SET OF CRITERIA(Petrozavodsk State University, 2020) Argyros, I.K.; George, S.Different set of criteria based on the seventh derivative are used for convergence of sixth order methods. Then, these methods are compared using numerical examples. But we do not know: if the results of those comparisons are true if the examples change; the largest radii of convergence; error estimates on distance between the iterate and solution, and uniqueness results that are computable. We address these concerns using only the first derivative and a common set of criteria. Numerical experiments are used to test the convergence criteria and further validate the theoretical results. Our technique can be used to make comparisons between other methods of the same order. © Petrozavodsk State University, 2020Item Extending the Convexity of Nonlinear Image of a Ball Appearing in Optimization(Universal Wiser Publisher, 2020) Argyros, I.K.; Cho, Y.J.; George, S.Let X, Y be Hilbert spaces and F: X ? Y be Fréchet differentiable. Suppose that F? is center-Lipschitz on U(w, r) and F?(w) be a surjection. Then, S1 = F(U(w, ?1)) is convex where ?1 ? r. The set S1 contains the corresponding set given in [18] under the Lipschitz condition. Numerical examples where the old conditions are not satisfied but the new conditions are satisfied are provided in this paper. © 2020 Ioannis K. Argyros, et al. DOI: https://doi.o.Item Extended Local Convergence for High Order Schemes Under ?-Continuity Conditions(Universal Wiser Publisher, 2020) Argyros, G.; Argyros, M.; Argyros, I.K.; George, S.There is a plethora of schemes of the same convergence order for generating a sequence approximating a solution of an equation involving Banach space operators. But the set of convergence criteria is not the same in general. This makes the comparison between them challenging and only numerically. Moreover, the convergence is established using Taylor series and by assuming the existence of high order derivatives that do not even appear on these schemes. Furthermore, no computable error estimates, uniqueness for the solution results or a ball of convergence is given. We address all these problems by using only the first derivative that actually appears on these schemes and under the same set of convergence conditions. Our technique is so general that it can be used to extend the applicability of other schemes along the same lines. © 2020 Ioannis K. Argyros, et al.