Faculty Publications
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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 Extended kung–traub methods for solving equations with applications(MDPI, 2021) Regmi, S.; Argyros, I.K.; George, S.; Magreñán Ruiz, Á.A.; Argyros, M.Kung and Traub (1974) proposed an iterative method for solving equations defined on the real line. The convergence order four was shown using Taylor expansions, requiring the existence of the fifth derivative not in this method. However, these hypotheses limit the utilization of it to functions that are at least five times differentiable, although the methods may converge. As far as we know, no semi-local convergence has been given in this setting. Our goal is to extend the applicability of this method in both the local and semi-local convergence case and in the more general setting of Banach space valued operators. Moreover, we use our idea of recurrent functions and conditions only on the first derivative and divided difference, which appear in the method. This idea can be used to extend other high convergence multipoint and multistep methods. Numerical experiments testing the convergence criteria complement this study. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item On the semi-local convergence of an ostrowski-type method for solving equations(MDPI, 2021) Argyros, C.I.; Argyros, I.K.; Joshi, J.; Regmi, S.; George, S.Symmetries play a crucial role in the dynamics of physical systems. As an example, microworld and quantum physics problems are modeled on principles of symmetry. These problems are then formulated as equations defined on suitable abstract spaces. Then, these equations can be solved using iterative methods. In this article, an Ostrowski-type method for solving equations in Banach space is extended. This is achieved by finding a stricter set than before containing the iterates. The convergence analysis becomes finer. Due to the general nature of our technique, it can be utilized to enlarge the utilization of other methods. Examples finish the paper. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item Convergence criteria of three step schemes for solving equations(MDPI, 2021) Regmi, S.; Argyros, C.I.; Argyros, I.K.; George, S.We develop a unified convergence analysis of three-step iterative schemes for solving nonlinear Banach space valued equations. The local convergence order has been shown before to be five on the finite dimensional Euclidean space assuming Taylor expansions and the existence of the sixth derivative not on these schemes. So, the usage of them is restricted six or higher differentiable mappings. But in our paper only the first Frèchet derivative is utilized to show convergence. Consequently, the scheme is expanded. Numerical applications are also given to test convergence. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Item Extended Kantorovich theory for solving nonlinear equations with applications(Springer Nature, 2023) Regmi, S.; Argyros, I.K.; George, S.; Argyros, M.The Kantorovich theory plays an important role in the study of nonlinear equations. It is used to establish the existence of a solution for an equation defined in an abstract space. The solution is usually determined by using an iterative process such as Newton’s or its variants. A plethora of convergence results are available based mainly on Lipschitz-like conditions on the derivatives, and the celebrated Kantorovich convergence criterion. But there are even simple real equations for which this criterion is not satisfied. Consequently, the applicability of the theory is limited. The question there arises: is it possible to extend this theory without adding convergence conditions? The answer is, Yes! This is the novelty and motivation for this paper. Other extensions include the determination of better information about the solution, i.e. its uniqueness ball; the ratio of quadratic convergence as well as more precise error analysis. The numerical section contains a Hammerstein-type nonlinear equation and other examples as applications. © 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.