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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.Item A procedure for increasing the convergence order of iterative methods from p to 5p for solving nonlinear system(Academic Press Inc., 2025) George, S.; M, M.; Gopal, M.; Godavarma, C.; Argyros, I.K.In this paper, we propose a procedure to obtain an iterative method that increases its convergence order from p to 5p for solving nonlinear systems. Our analysis is given in more general Banach space settings and uses assumptions on the derivative of the involved operator only up to order max?{k,2}. Here, k is the order of the highest derivative used in the convergence analysis of the iterative method with convergence order p. A particular case of our analysis includes an existing fifth-order method and improves its applicability to more problems than the problems covered by the method's analysis in earlier study. © 2024Item Improved convergence analysis for an at least fourth and at least sixth order parametric family of iterative methods for nonlinear system(Springer-Verlag Italia s.r.l., 2025) George, S.; Gopal, M.; M, M.Hueso et al. (2015) introduced a new family of iterative methods for solving non-linear systems. However, the convergence analysis is based on Taylor series expansion, which requires the existence of derivatives of the involved operator up to the fifth and seventh orders, respectively, for the method with at least fourth-order convergence and the method with sixth-order convergence. In this paper, we obtain at least fourth- and sixth-order convergence for the respective methods by assuming derivatives only up to the third order. We also provide the semi-local convergence analysis (which is not given in Hueso et al. (2015)) in a more general Banach space form. Moreover, our semi-local and local convergence analyses are based on the same set of assumptions, unlike existing studies, where the authors typically use one set of assumptions for semi-local convergence analysis and another set of assumptions for local convergence analysis. Numerical examples and dynamics of the methods are also provided in this study. © The Author(s), under exclusive licence to Springer-Verlag Italia S.r.l., part of Springer Nature 2025.Item An Improved Convergence Analysis of a Multi-Step Method with High-Efficiency Indices(Multidisciplinary Digital Publishing Institute (MDPI), 2025) George, S.; Gopal, M.; Bhide, S.; Argyros, I.K.A multi-step method introduced by Raziyeh and Masoud for solving nonlinear systems with convergence order five has been considered in this paper. The convergence of the method was studied using Taylor series expansion, which requires the function to be six times differentiable. However, our convergence study does not depend on the Taylor series. We use the derivative of F up to two only in our convergence analysis, which is presented in a more general Banach space setting. Semi-local analysis is also discussed, which was not given in earlier studies. Unlike in earlier studies (where two sets of assumptions were used), we used the same set of assumptions for semi-local analysis and local convergence analysis. We discussed the dynamics of the method and also gave some numerical examples to illustrate theoretical findings. © 2025 by the authors.