Faculty Publications
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Item Expanding the applicability of a modified Gauss-Newton method for solving nonlinear ill-posed problems(2013) Argyros, I.K.; George, S.We expand the applicability of a modified Gauss-Newton method recently presented in George (2013) [19] for approximate solution of a nonlinear ill-posed operator equation between two Hilbert spaces. We use a center-type Lipschitz condition in our convergence analysis instead of a Lipschitz-type condition used in earlier studies such as George (2013, 2010) [19,18]. This way a tighter convergence analysis is obtained and under less computational cost, since the more precise and easier to compute center-Lipschitz instead of the Lipschitz constant is used in the convergence analysis. Numerical examples are presented to show that our results apply but earlier ones do not apply to solve equations. © 2013 Elsevier Inc. All rights reserved.Item Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions(Springer-Verlag Italia s.r.l., 2016) Argyros, I.K.; George, S.We present a local convergence analysis of a sixth order iterative method for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Sharma et al. (Appl Math Comput 190:111–115, 2007) have shown convergence of these methods under hypotheses up to the fifth derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study. © 2015, Springer-Verlag Italia.Item On the convergence of Broyden's method with regularity continuous divided differences and restricted convergence domains(Mathematical Research Press, 2017) Argyros, I.K.; George, S.We present a semilocal convergence analysis for Broyden's method with regularly continuous divided differences in a Banach/Hilbert space setting. By using: center-Lipschitz-type conditions defining restricted convergence domains, at least as weak hypotheses and the same computational cost as in [6] we provide a new convergence analysis for Broyden's method with the following advantages: larger convergence domain; finer error bounds on the distances involved, and at least as precise information on the location of the solution. © 2017 Journal of Nonlinear Functional Analysis.Item Unified convergence analysis of frozen Newton-like methods under generalized conditions(Elsevier B.V., 2019) Argyros, I.K.; George, S.The objective in this article is to present a unified convergence analysis of frozen Newton-like methods under generalized Lipschitz-type conditions for Banach space valued operators. We also use our new idea of restricted convergence domains, where we find a more precise location, where the iterates lie leading to at least as tight majorizing functions. Consequently, the new convergence criteria are weaker than in earlier works resulting to the expansion of the applicability of these methods. The conditions do not necessarily imply the differentiability of the operator involved. This way our method is suitable for solving equations and systems of equations. Numerical examples complete the presentation of this article. © 2018 Elsevier B.V.Item On the complexity of choosing majorizing sequences for iterative procedures(Springer-Verlag Italia s.r.l., 2019) Argyros, I.K.; George, S.The aim of this paper is to introduce general majorizing sequences for iterative procedures which may contain a non-differentiable operator in order to solve nonlinear equations involving Banach valued operators. A general semi-local convergence analysis is presented based on majorizing sequences. The convergence criteria, if specialized can be used to study the convergence of numerous procedures such as Picard’s, Newton’s, Newton-type, Stirling’s, Secant, Secant-type, Steffensen’s, Aitken’s, Kurchatov’s and other procedures. The convergence criteria are flexible enough, so if specialized are weaker than the criteria given by the aforementioned procedures. Moreover, the convergence analysis is at least as tight. Furthermore, these advantages are obtained using Lipschitz constants that are least as tight as the ones already used in the literature. Consequently, no additional hypotheses are needed, since the new constants are special cases of the old constants. These ideas can be used to study, the local convergence, multi-step multi-point procedures along the same lines. Some applications are also provided in this study. © 2018, Springer-Verlag Italia S.r.l., part of Springer Nature.Item Local convergence analysis of jarratt-type schemes for solving equations(Biemdas Academic Publishers, 2019) Argyros, I.K.; George, S.The aim of this paper is to present a local convergence analysis for Jarratt-type schemes to solve equations with Banach space valued operators. The convergence conditions are are weaker than the corresponding ones in earlier results and some numerical examples are provided. © 2019 Applied Set-Valued Analysis and Optimization.Item Expanding the applicability of an a posteriori parameter choice strategy for Tikhonov regularization of nonlinear ill-posed problems(Springer-Verlag Italia s.r.l., 2019) Argyros, I.K.; Cho, Y.J.; George, S.; Xiao, Y.We expand the applicability of an a posteriori parameter choice strategy for Tikhonov regularization of the nonlinear ill-posed problem presented in Jin and Hou (Numer Math 83:139–159, 1999) by weakening the conditions needed in Jin and Hou [13]. Using a center-type Lipschitz condition instead of a Lipschitz-type condition used in Jin and Hou [13], Scherzer et al. (SIAM J Numer Anal 30:1796–1838, 1993), we obtain a tighter convergence analysis. Numerical examples are presented to show that our results apply but earlier ones do not apply to solve equations. © 2019, The Royal Academy of Sciences, Madrid.Item Convergence analysis for single point Newton-type iterative schemes(Springer, 2020) Argyros, I.K.; George, S.The aim of this article is to present a convergence analysis for single point Newton-type schemes for solving equations with Banach space valued operators. The equations contain a non-differentiable part. Although the convergence conditions are very general, they are weaker than the corresponding ones in earlier works leading to a finer convergence analysis in both the local as well as the semi-local convergence analysis. Therefore, the applicability of these iterative schemes is extended. © 2019, Korean Society for Informatics and Computational Applied Mathematics.Item On the complexity of extending the convergence region for Traub's method(Academic Press Inc. apjcs@harcourt.com, 2020) Argyros, I.K.; George, S.The convergence region of Traub's method for solving equations is small in general. This fact limits its applicability. We locate a more precise region containing the Traub iterations leading to at least as tight Lipschitz constants as before. Our convergence analysis is finer, and obtained without additional conditions. The new theoretical results are tested on numerical examples that illustrate their superiority over earlier results. © 2019Item Extended domain for fifth convergence order schemes(Universidad de la Frontera, 2021) Argyros, I.K.; George, S.We provide a local as well as a semi-local analysis of a fifth convergence order scheme involving operators valued on Banach space for solving nonlinear equations. The convergence domain is extended resulting a finer convergence analysis for both types. This is achieved by locating a smaller domain included in the older domain leading this way to tighter Lipschitz type functions. These extensions are obtained without additional hypotheses. Numerical examples are used to test the convergence criteria and also to show the superiority for our results over earlier ones. Our idea can be utilized to extend other schemes using inverses in a similar way. © 2021 I. K. Argyros et al.