Faculty Publications
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Item Enhancing the practicality of Newton–Cotes iterative method(Institute for Ionics, 2023) Sadananda, R.; George, S.; Kunnarath, A.; Padikkal, J.; Argyros, I.K.The new Newton-type iterative method developed by Khirallah et al. (Bull Math Sci Appl 2:01–14, 2012), is shown to be of the convergence order three, without the application of Taylor series expansion. Our analysis is based on the assumptions on second order derivative of the involved operator, unlike the earlier studies. Moreover, this technique is extended to methods of higher order of convergence, five and six. This paper also verifies the theoretical approach using numerical examples and comparisons, in addition to the visualization of Julia and Fatou sets of the corresponding methods. © 2023, The Author(s) under exclusive licence to Korean Society for Informatics and Computational Applied Mathematics.Item On the convergence of open Newton’s method(Springer Science and Business Media B.V., 2023) Kunnarath, A.; George, S.; Sadananda, R.; Padikkal, J.; Argyros, I.K.Cordero and Torregrosa proved the convergence of two Newton’s-like methods in 2007. Using Taylor expansion (requiring existence of derivatives of order up to four of the involved operator) they obtained the convergence order three for these methods. The convergence order three is obtained for Open Newton’s method and two extensions of it with assumptions only on first two derivatives of the operator involved. We verified the results with examples and dynamics of the results are presented. © 2023, The Author(s), under exclusive licence to The Forum D’Analystes.Item On obtaining order of convergence of Jarratt-like method without using Taylor series expansion(Springer Nature, 2024) George, S.; Kunnarath, A.; Sadananda, R.; Padikkal, J.; Argyros, I.K.In 2014, Sharma and Arora introduced two efficient Jarratt-like methods for solving systems of non-linear equations which are of convergence order four and six. To prove the respective convergence order, they used Taylor expansion which demands existence of derivative of the function up to order seven. In this paper, we obtain the respective convergence order for these methods using assumptions only on first three derivatives of the function. Other problems with this approach are: the lack of computable a priori estimates on the error distances involved as well as isolation of the solution results. These concerns constitute our motivation for this article. One extension of the fourth order method is presented which is of convergence order eight and the same is proved without any extra assumptions on the higher order derivatives. All the results are proved in a general Banach space setting. Numerical examples and dynamics of the methods are studied to analyse the performance of the method. © The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional 2024.Item Unified Convergence Analysis of Certain At Least Fifth Order Methods(SINUS Association, 2025) Sadananda, R.; Gopal, M.; George, S.; Argyros, I.K.A class of iterative methods was developed by Xiao and Yin in 2015 and obtained convergence order five using Taylor expansion. They had imposed the conditions on the derivatives of the involved operator of order at least up to four. In this paper, the order of convergence is achieved by imposing conditions only on the first two derivatives of the operator involved. The assumptions under consideration are weaker and the analysis is done in the more general setting of Banach spaces without using Taylor series expansion. The semi-local convergence analysis is also given. Further, the theory is justified by numerical examples. © 2024, SINUS Association. All rights reserved.