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Item Unified semi-local convergence for k-Step iterative methods with flexible and frozen linear operator(2018) Argyros, I.K.; George, S.The aim of this article is to present a unified semi-local convergence analysis for a k-step iterative method containing the inverse of a flexible and frozen linear operator for Banach space valued operators. Special choices of the linear operator reduce the method to the Newton-type, Newton's, or Stirling's, or Steffensen's, or other methods. The analysis is based on center, as well as Lipschitz conditions and our idea of the restricted convergence region. This idea defines an at least as small region containing the iterates as before and consequently also a tighter convergence analysis. � 2018 by the authors.Item Unified convergence analysis of frozen Newton-like methods under generalized conditions(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 Unified Convergence for Multi-Point Super Halley-Type Methods with Parameters in Banach Space(2019) Argyros, I.K.; George, S.We present a local convergence analysis of a multi-point super-Halley-like method in order to approximate a locally unique solution of an equation in a Banach space setting. The convergence analysis in earlier works was based on hypotheses reaching up to the third derivative of the operator. In the present study we expand the applicability of the Super-Halley-like method by using hypotheses only on the first derivative. We also provide: A computable error on the distances involved and a uniqueness result based on Lipschitz constants. The convergence order is also provided for these methods. Numerical examples are also presented in this study. � 2019, Indian National Science Academy.Item Third-order derivative-free methods in Banach spaces for nonlinear ill-posed equations(2019) Shubha, V.S.; George, S.; Jidesh, P.We develop three third order derivative-free iterative methods to solve the nonlinear ill-posed oprerator equation F(x) = f approximately. The methods involve two steps and are free of derivatives. Convergence analysis shows that these methods converge cubically. The adaptive scheme introduced in Pereverzyev and Schock (SIAM J Numer Anal 43(5):2060 2076, 2005) has been employed to choose regularization parameter. These methods are applied to the inverse gravimetry problem to validate our developed results. 2019, Korean Society for Computational and Applied Mathematics.Item On the complexity of extending the convergence region for Traub's method(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 On the complexity of choosing majorizing sequences for iterative procedures(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 On a Two-Step Kurchatov-Type Method in Banach Space(2019) Argyros, I.K.; George, S.We present the semi-local convergence analysis of a two-step Kurchatov-type method to solve equations involving Banach space valued operators. The analysis is based on our ideas of recurrent functions and restricted convergence region. The study is completed using numerical examples. 2019, Springer Nature Switzerland AG.Item Numerical approximation of a Tikhonov type regularizer by a discretized frozen steepest descent method(2018) George, S.; Sabari, M.We present a frozen regularized steepest descent method and its finite dimensional realization for obtaining an approximate solution for the nonlinear ill-posed operator equation F(x)=y. The proposed method is a modified form of the method considered by Argyros et al. (2014). The balancing principle considered by Pereverzev and Schock (2005) is used for choosing the regularization parameter. The error estimate is derived under a general source condition and is of optimal order. The provided numerical example proves the efficiency of the proposed method. 2017 Elsevier B.V.Item Modified Minimal Error Method for Nonlinear Ill-Posed Problems(2018) Sabari, M.; George, S.An error estimate for the minimal error method for nonlinear ill-posed problems under general a H lder-type source condition is not known. We consider a modified minimal error method for nonlinear ill-posed problems. Using a H lder-type source condition, we obtain an optimal order error estimate. We also consider the modified minimal error method with noisy data and provide an error estimate. 2018 Walter de Gruyter GmbH, Berlin/Boston.Item Local Convergence of Inexact Newton-Like Method under Weak Lipschitz Conditions(2020) Argyros, I.K.; Cho, Y.J.; George, S.; Xiao, Y.The paper develops the local convergence of Inexact Newton-Like Method (INLM) for approximating solutions of nonlinear equations in Banach space setting. We employ weak Lipschitz and center-weak Lipschitz conditions to perform the error analysis. The obtained results compare favorably with earlier ones such as [7, 13, 14, 18, 19]. A numerical example is also provided. 2020, Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences.