Faculty Publications
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Item Local convergence for some high convergence order Newton-like methods with frozen derivatives(Springer Nature, 2015) Argyros, I.K.; George, S.We present a local convergence analysis of some families of Newton-like methods with frozen derivatives in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Amat et al. (Appl Math Lett. 25:2209–2217, 2012), Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam, 2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Xiao and Yin (Appl Math Comput, 2015) the local convergence was proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these methods. In this paper we expand the applicability of these methods using only hypotheses on the first derivative and Lipschitz constants. Numerical examples are also presented in this study. © 2015, Sociedad Española de Matemática Aplicada.Item On a sixth-order Jarratt-type method in Banach spaces(World Scientific, 2015) Argyros, I.K.; George, S.We present a local convergence analysis of a sixth-order Jarratt-type method in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies such as [X. Wang, J. Kou and C. Gu, Semilocal convergence of a sixth-order Jarratt method in Banach spaces, Numer. Algorithms 57 (2011) 441-456.] require hypotheses up to the third Fréchet-derivative. Numerical examples are also provided in this study. © 2015 World Scientific Publishing Company.Item Local convergence of deformed Jarratt-type methods in Banach space without inverses(World Scientific Publishing Co. Pte Ltd wspc@wspc.com.sg, 2016) Argyros, I.K.; George, S.We present a local convergence analysis for the Jarratt-type method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet-derivative. Hence, the applicability of these methods is expanded under weaker hypotheses and less computational cost for the constants involved in the convergence analysis. Numerical examples are also provided in this study. © World Scientific Publishing Company.Item Local convergence of deformed Euler-Halley-type methods in Banach space under weak conditions(World Scientific Publishing Co. Pte Ltd wspc@wspc.com.sg, 2017) Argyros, I.K.; George, S.We present a unified local convergence analysis for deformed Euler-Halley-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Euler, Halley and other high order methods. The convergence ball and error estimates are given for these methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second Fréchet derivative. Numerical examples are also provided in this study. © 2017 World Scientific Publishing Company.Item EXTENDING THE RADIUS OF CONVERGENCE FOR A CLASS OF EULER-HALLEY TYPE METHODS(Publishing House of the Romanian Academy, 2019) Argyros, I.K.; George, S.The aim of this paper is to extend the radius of convergence and improve the ratio of convergence for a certain class of Euler-Halley type methods with one parameter in a Banach space. These improvements over earlier works are obtained using the same functions as before but more precise information on the location of the iterates. Special cases and examples are also presented in this study. © 2019, Publishing House of the Romanian Academy. All rights reserved.Item Local convergence of an at least sixth-order method in Banach spaces(Birkhauser Verlag AG, 2019) Argyros, I.K.; Khattri, S.K.; George, S.We present a local convergence analysis of an at least sixth-order family of methods to approximate a locally unique solution of nonlinear equations in a Banach space setting. The semilocal convergence analysis of this method was studied by Amat et al. in (Appl Math Comput 206:164–174, 2008; Appl Numer Math 62:833–841, 2012). This work provides computable convergence ball and computable error bounds. Numerical examples are also provided in this study. © 2019, Springer Nature Switzerland AG.Item Unified Convergence for Multi-Point Super Halley-Type Methods with Parameters in Banach Space(Indian National Science Academy, 2019) Argyros, I.K.; George, S.We present a local convergence analysis of a multi-point super-Halley-like method in order to approximate a locally unique solution of an equation in a Banach space setting. The convergence analysis in earlier works was based on hypotheses reaching up to the third derivative of the operator. In the present study we expand the applicability of the Super-Halley-like method by using hypotheses only on the first derivative. We also provide: A computable error on the distances involved and a uniqueness result based on Lipschitz constants. The convergence order is also provided for these methods. Numerical examples are also presented in this study. © 2019, Indian National Science Academy.Item Extending the Applicability of the Super-Halley-Like Method Using ?-Continuous Derivatives and Restricted Convergence Domains(Sciendo, 2019) Argyros, I.K.; George, S.We present a local convergence analysis of the super-Halley-like method in order to approximate a locally unique solution of an equation in a Banach space setting. The convergence analysis in earlier studies was based on hypotheses reaching up to the third derivative of the operator. In the present study we expand the applicability of the super-Halley-like method by using hypotheses up to the second derivative. We also provide: a computable error on the distances involved and a uniqueness result based on Lipschitz constants. Numerical examples are also presented in this study. © 2019 Ioannis K. Argyros et al., published by Sciendo.Item 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, 2020