Faculty Publications
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Item Local convergence of deformed Halley method in Banach space under Holder continuity conditions(International Scientific Research Publications editorial-office@tjnsa.com, 2015) Argyros, I.K.; George, S.We present a local convergence analysis for deformed Halley method in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Halley and other high order methods under hypotheses up to the first Fréchet-derivative in contrast to earlier studies using hypotheses up to the second or third Fréchet-derivative. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study. © 2015, International Scientific Research Publications. All rights reserved.Item Ball Convergence for two-parameter chebyshev-halley-like methods in banach space using hypotheses only on the first derivative(International Publications internationalpubls@yahoo.com, 2017) Argyros, I.K.; George, S.; Verma, R.U.We present a local convergence analysis of a sixth-order method for approximate a locally unique solution of an equation in the Banach space setting. The convergence of this methods is shown in Narang et al. (2016) under hypotheses up to the fourth Fréchet-derivative and the Lipschitz continuity of the third derivative, although only the first derivative appears in the method. In this study we expand the applicability of this method using only hypotheses on the first derivative of the function. Numerical examples are also presented in this study.Item Convergence analysis for a fast class of multi-step chebyshe-halley-type methods under weak conditions(International Publications internationalpubls@yahoo.com, 2020) Argyros, I.K.; George, S.In this study a convergence analysis for a fast multi-step Chebyshe-Halley-type method for solving nonlinear equations involving Banach space valued operator is presented. We introduce a more precise convergence region containing the iterates leading to tighter Lipschitz constants and functions. This way advantages are obtained in both the local as well as the semi-local convergence case under the same computational cost such as: extended convergence domain, tighter error bounds on the distances involved and a more precise information on the location of the solution. The new technique can be used to extend the applicability of other iterative methods. The numerical examples further validate the theoretical results. © 2020, International Publications. All rights reserved.Item Convergence analysis for a fast class of multi-step chebyshev-halley-type methods under weak conditions(International Publications internationalpubls@yahoo.com, 2020) Argyros, I.K.; George, S.In this study a convergence analysis for a fast multi-step Chebyshe-Halley-type method for solving nonlinear equations involving Banach space valued operator is presented. We introduce a more precise convergence region containing the iterates leading to tighter Lipschitz constants and functions. This way advantages are obtained in both the local as well as the semi-local convergence case under the same computational cost such as: extended convergence domain, tighter error bounds on the distances involved and a more precise in-formation on the location of the solution. The new technique can be used to extend the applicability of other iterative methods. The numerical examples further validate the theoretical results. © 2020, International Publications. All rights reserved.Item Improving the radius of convergence for the traub’s method for multiple roots(International Publications internationalpubls@yahoo.com, 2020) Argyros, I.K.; George, S.The aim of this paper is to find the radius of convergence of Traub’s method for solving nonlinear equations with roots of multiplicity greater or equal to one under conditions more general than in earlier studies. This way we expand the applicability of Traub’s method. © 2020, International Publications. All rights reserved.Item Hybrid Chebyshev-Type Methods for Solving Nonlinear Equations(Multidisciplinary Digital Publishing Institute (MDPI), 2025) Argyros, I.K.; George, S.Chebyshev-type methods have replaced the Chebyshev method in practice for solving nonlinear equations in abstract spaces. These methods are of the same R-order of three. However, they are easier to deal with, since the computationally expensive second derivative of the operator involved does not appear on these methods. However, the invertibility of the first derivative is still required at each step of the iteration. In this article, the inverse is replaced by a finite sum of linear operators. The convergence of the new Hybrid Chebyshev-Type Method (HCTM) is established under relaxed generalized continuity assumptions on the derivative and majorizing sequences. The iterates of the new methods converge to the original ones, but they are easier to find. Moreover, the numerical examples demonstrate that the new iterates converge essentially as fast to the solution. The methodology of this article can be used on other methods with inverses along the same lines due to its generality. © 2024 by the authors.