1. Journal Articles
Permanent URI for this collectionhttps://idr.nitk.ac.in/handle/1/6
Browse
52 results
Search Results
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 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.Item 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 Local results for an iterative method of convergence order six and efficiency index 1.8171(2017) Argyros, I.K.; George, S.We present a local convergence analysis of an iterative method of convergence order six and efficiency index 1.8171 in order to approximate a locally unique solution of a nonlinear equation. In earlier studies such as [16] the convergence order of these methods was given under hypotheses reaching up to the fourth derivative of the function although only the first derivative appears in these methods. In this paper, we expand the applicability of these methods by showing convergence using only the first and second derivatives. Moreover, we compare the convergence radii and provide computable error estimates for these methods using Lipschitz constants. 2017, Institute of Mathematics. All rights reserved.