Faculty Publications

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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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    Convergence Analysis of Jarratt-like Methods for Solving Nonlinear Equations for Thrice-Differentiable Operators
    (Multidisciplinary Digital Publishing Institute (MDPI), 2025) Bate, I.; Senapati, K.; George, S.; Argyros, I.K.; Argyros, M.I.
    The main goal of this paper is to study Jarratt-like iterative methods to obtain their order of convergence under weaker conditions. Generally, obtaining the (Formula presented.) -order convergence using the Taylor series expansion technique needed at least (Formula presented.) times differentiability of the involved operator. However, we obtain the fourth- and sixth-order for Jarratt-like methods using up to the third-order derivatives only. An upper bound for the asymptotic error constant (AEC) and a convergence ball are provided. The convergence analysis is developed in the more general setting of Banach spaces and relies on Lipschitz-type conditions, which are required to control the derivative. The results obtained are examined using numerical examples, and some dynamical system concepts are discussed for a better understanding of convergence ideas. © 2025 by the authors.