Faculty Publications
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Publications by NITK Faculty
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Item Ball Convergence for Second Derivative Free Methods in Banach Space(Springer, 2017) Argyros, I.K.; Padikkal, P.; George, S.Hueso et al. (Appl Math Comput 211:190–197, 2009) considered a third and fourth order iterative methods for nonlinear systems. The methods were shown to of order third and fourth if the operator equation is defined on the j-dimensional Euclidean space (Hueso et al. in Appl Math Comput 211:190–197, 2009). The order of convergence was shown using hypotheses up to the third Fréchet derivative of the operator involved although only the first derivative appears in these methods. In the present study we only use hypotheses on the first Fréchet-derivative. This way the applicability of these methods is expanded. Moreover we present a radius of convergence a uniqueness result and computable error bounds based on Lipschitz constants. Numerical examples are also presented in this study. © 2015, Springer India Pvt. Ltd.Item Convergence Analysis of a Fifth-Order Iterative Method Using Recurrence Relations and Conditions on the First Derivative(Birkhauser, 2021) George, S.; Argyros, I.K.; Padikkal, P.; Mahapatra, M.; Saeed, M.Using conditions on the second Fréchet derivative, fifth order of convergence was established in Singh et al. (Mediterr J Math 13:4219–4235, 2016) for an iterative method. In this paper, we establish fifth order of convergence of the method using assumptions only on the first Fréchet derivative of the involved operator. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.Item Local Convergence of Traub’s Method and Its Extensions(MDPI, 2023) Saeed K, M.; Remesh, K.; George, S.; Padikkal, P.; Argyros, I.K.In this article, we examine the local convergence analysis of an extension of Newton’s method in a Banach space setting. Traub introduced the method (also known as the Arithmetic-Mean Newton’s Method and Weerakoon and Fernando method) with an order of convergence of three. All the previous works either used higher-order Taylor series expansion or could not derive the desired order of convergence. We studied the local convergence of Traub’s method and two of its modifications and obtained the convergence order for these methods without using Taylor series expansion. The radii of convergence, basins of attraction, comparison of iterations of similar iterative methods, approximate computational order of convergence (ACOC), and a representation of the number of iterations are provided. © 2023 by the authors.Item On the Order of Convergence and the Dynamics of Werner-King’s Method(Universal Wiser Publisher, 2023) George, S.; Argyros, I.K.; Kunnarath, A.; Padikkal, P.In this paper, we present the local convergence analysis of Werner-King’s method to approximate the solution of a nonlinear equation in Banach spaces. We establish the local convergence theorem under conditions on the first and second Fréchet derivatives of the operator involved. The convergence analysis is not based on the Taylor expansions as in the earlier studies (which require the assumptions on the third order Fréchet derivative of the operator involved). Thus our analysis extends the applicability of Werner-King’s method. We illustrate our results with numerical examples. Moreover, the dynamics and the basins of attraction are developed and demonstrated. © 2023 Santhosh George, et al.