Faculty Publications

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Author: search.filters.author.Padikkal, P.Subject: search.filters.subject.order of convergence

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    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.
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    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.