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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 Shock coupled fourth-order diffusion for image enhancement(2012) Jidesh, P.; George, S.In this paper a shock coupled fourth-order diffusion filter is proposed for image enhancement. This filter converges at a faster rate while preserving and enhancing edges, ramps and textures present in the images. The proposed filter diffuses with varying magnitudes in the directions normal to the level-curve and along it. The magnitude of the directional diffusion is controlled by a diffusion function, meant to provide a good response in the direction along the level-curves, than across them. The proposed filter can still preserve the planar approximation of the image, thereby avoiding the discrepancy caused due to the staircase effect, as in the second-order counterparts. The anisotropic property of the filter is thoroughly studied, analyzed and demonstrated with perspective and quantitative results. The performance of the proposed filter is compared with the state-of-the-art methods for image enhancement. The quantitative and perspective measures provided endorse the capability of the method to enhance various kinds of images. 2012 Elsevier Ltd. All rights reserved.Item Reconstruction of signals by standard Tikhonov method(2011) George, S.; Jidesh, P.In this work we propose a standard Tikhonov regularization approach for obtaining the signal f from the observed signal ye. The observed signal is distorted by an additive noise or error e. Deviating from the usual assumption on the bound onItem Proximal methods with invexity and fractional calculus(2017) Anastassiou, G.A.; Argyros, I.K.; George, S.We present some proximal methods with invexity results involving fractional calculus.Item On the local convergence of newton-like methods with fourth and fifth order of convergence under hypotheses only on the first fr chet derivative(2017) Argyros, I.K.; Jidesh, P.; George, S.We present a local convergence analysis of several Newton-like methods with fourth and fifth order of convergence in order to approximate a locally unique solution of an equation in Banach space setting. Earlier studies have used hypotheses up to the fifth derivative although only the first derivative appears in the definition of these methods. In this study we only use the hypothesis on the first derivative. This way we expand the applicability of these methods. Moreover, we provide a radius of convergence, a uniqueness ball and computable error bounds based on Lipschitz constants. Numerical examples computing the radii of the convergence balls as well as examples where earlier results cannot apply to solve equations but our results can apply are also given in this study. 2017, Institute of Mathematics. All rights reserved.Item On the Convergence of Stirling s Method for Fixed Points Under Not Necessarily Contractive Hypotheses(2017) Argyros, I.K.; Muruster, S.; George, S.Stirling s method is a useful alternative to Newton s method for approximating fixed points of nonlinear operators in a Banach space setting. This method has been studied under contractive hypotheses on the operator involved, thus limiting the applicability of it. In this study, we present a local as well as a semi-local convergence for this method based on not necessarily contractive hypotheses. This way, we extend the applicability of the method. Moreover, we present a favorable comparison of the new Kantorovich-type convergence criteria with the old ones using contractive hypotheses as well as with Newton s method. Numerical examples including Hammerstein nonlinear equations of Chandrasekar type appearing in neutron transport and in the kinetic theory of gases are solved to further illustrate the theoretical results. 2017, Springer (India) Private Ltd.Item On the convergence of Newton-like methods using restricted domains(2017) Argyros, I.K.; George, S.We present a new semi-local convergence analysis for Newton-like methods in order to approximate a locally unique solution of a nonlinear equation containing a non-differentiable term in a Banach space setting. The new idea uses more precise convergence domains. This way the new sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies. Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar-type, are also provided in this study. 2016, Springer Science+Business Media New York.