Faculty Publications
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Item On an eighth order steffensen-type solver free of derivatives(Nova Science Publishers, Inc., 2019) Argyros, I.K.; George, S.We expand the applicability of an eighth convergence order Steffensen-type solver for equations involvingBanach space valued operators using only the first order derivative in contrast to earlier works using derivatives of order five which do not appear in the method, and in the special case of the i-dimensional Euclidean space. © 2020 by Nova Science Publishers, Inc. All rights reserved.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 Local comparison between two ninth convergence order algorithms for equations(MDPI AG rasetti@mdpi.com Postfach Basel CH-4005, 2020) Regmi, S.; Argyros, I.K.; George, S.A local convergence comparison is presented between two ninth order algorithms for solving nonlinear equations. In earlier studies derivatives not appearing on the algorithms up to the 10th order were utilized to show convergence. Moreover, no error estimates, radius of convergence or results on the uniqueness of the solution that can be computed were given. The novelty of our study is that we address all these concerns by using only the first derivative which actually appears on these algorithms. That is how to extend the applicability of these algorithms. Our technique provides a direct comparison between these algorithms under the same set of convergence criteria. This technique can be used on other algorithms. Numerical experiments are utilized to test the convergence criteria. © 2020 by the authors.Item Convergence analysis for a fast class of multi-step chebyshe-halley-type methods under weak conditions(International Publications internationalpubls@yahoo.com, 2020) Argyros, I.K.; George, S.In this study a convergence analysis for a fast multi-step Chebyshe-Halley-type method for solving nonlinear equations involving Banach space valued operator is presented. We introduce a more precise convergence region containing the iterates leading to tighter Lipschitz constants and functions. This way advantages are obtained in both the local as well as the semi-local convergence case under the same computational cost such as: extended convergence domain, tighter error bounds on the distances involved and a more precise information on the location of the solution. The new technique can be used to extend the applicability of other iterative methods. The numerical examples further validate the theoretical results. © 2020, International Publications. All rights reserved.Item Convergence analysis for a fast class of multi-step chebyshev-halley-type methods under weak conditions(International Publications internationalpubls@yahoo.com, 2020) Argyros, I.K.; George, S.In this study a convergence analysis for a fast multi-step Chebyshe-Halley-type method for solving nonlinear equations involving Banach space valued operator is presented. We introduce a more precise convergence region containing the iterates leading to tighter Lipschitz constants and functions. This way advantages are obtained in both the local as well as the semi-local convergence case under the same computational cost such as: extended convergence domain, tighter error bounds on the distances involved and a more precise in-formation on the location of the solution. The new technique can be used to extend the applicability of other iterative methods. The numerical examples further validate the theoretical results. © 2020, International Publications. All rights reserved.Item Local comparison between two-step methods under the same conditions(Springer Science and Business Media Deutschland GmbH, 2021) Argyros, I.K.; George, S.In earlier studies different methods of same convergence order are campared using numerical examples. The drawback of this approach is that, 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. In this paper we campare the ball convergence of two-step iterative methods for solving the equation G(x) = 0 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. © 2021, African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature.Item On the local convergence of two novel schemes of convergence order eight for solving equations: An extension(International Publications, 2021) Argyros, I.K.; George, S.; Argyros, C.I.We extend the applicability of two eighth order schemes for solving nonlinear equations for Banach space valued equations.This is done by using assumptions only on the first derivative that does appear on the schemes, whereas in earlier works up to the ninth derivative (not on the scheme) are used to establish the convergence. Our technique is so general that it can be used to extend the usage of other schemes along the same lines. © 2021, International Publications. All rights reserved.Item On the convergence of Homeier method and its extensions(Springer Science and Business Media B.V., 2022) Muhammed Saeed, K.; Krishnendu, R.; George, S.; Padikkal, J.A third-order Homeier method for solving equations in Banach space is studied. Using assumptions on the first and second derivatives, we obtained third-order convergence. Our technique does not involve Taylor series expansion and can be extended to similar higher-order methods. We have given two extensions of the method with orders five and six. Examples with radii of convergence and basins of attraction are provided. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.Item Advances in Nonlinear Variational Inequalities Volume 25 (2022), Number 1, 49-58 Comparing and Extending Two Fourth Order Methods Under the Same Hypotheses for Equations(International Publications, 2022) Argyros, I.K.; George, S.; Argyros, C.I.We compare and extend two fourth order methods for nonlinear equations. Our convergence analysis used assumptions only on the first derivative. Earlier studies have used hypotheses up to the fifth derivative, limiting the applicability of the method. Numerical examples complete the article. © 2022, International Publications. All rights reserved.Item On the convergence of a novel seventh convergence order schemes for solving equations(Springer Science and Business Media B.V., 2022) Regmi, S.; Argyros, I.K.; George, S.; Argyros, C.We study the local convergence of a seventh order scheme for solving nonlinear equations for Banach space valued equations. This is done by using assumptions only on the first derivative that does appear on the schemes, whereas in earlier works up to the eighth derivative (not on the scheme) are used to establish the convergence (not on the scheme). Our technique is so general that it can be used to extend the usage of other schemes along the same lines. © 2022, The Author(s), under exclusive licence to The Forum D’Analystes.