1. Journal Articles
Permanent URI for this collectionhttps://idr.nitk.ac.in/handle/1/6
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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 Unified convergence domains of Newton-like methods for solving operator equations(2016) Argyros, I.K.; George, S.We present a unified semilocal convergence analysis in order to approximate a locally unique zero of an operator equation in a Banach space setting. Using our new idea of restricted convergence domains we generate smaller Lipschitz constants than in earlier studies leading to the following advantages: weaker sufficient convergence criteria, tighter error estimates on the distances involved and an at least as precise information on the location of the zero. Hence, the applicability of these methods is extended. These advantages are obtained under the same cost on the parameters involved. Numerical examples where the old sufficient convergence criteria cannot apply to solve equations but the new criteria can apply are also provided in this study. � 2016 Elsevier Inc. All rights reserved.Item The asymptotic mesh independence principle of Newton's method under weaker conditions(2016) Argyros, I.K.; Sheth, S.M.; Younis, R.M.; George, S.We present a new asymptotic mesh independence principle of Newton's method for discretized nonlinear operator equations. Our hypotheses are weaker than in earlier studies such as [1], [8]-[12]. This way we extend the applicability of the mesh independence principle which asserts that the behavior of the discretized version is asymptotically the same as that of the original iteration and consequently, the number of steps required by the two processes to converge within a given tolerance is essentially the same. The results apply to solve a boundary value problem that cannot be solved with the earlier hypotheses given in [12]. 2016 International Publications. All rights reserved.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 a secant like method in a banach space under weak conditions(2016) Argyros, I.K.; Khattri, S.K.; George, S.We present a local convergence analysis of a Secant-like method in a Banach space setting. The method is used to approximate a solution of a nonlinear equation. The sufficient convergence conditions are weaker than in earlier studies. Numerical examples are also given in this work. 2016 International Publications. All rights reserved.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 semilocal convergence of newton's method for sections on riemannian manifolds(2014) Argyros, I.K.; George, S.; Dass, B.K.We present a semilocal convergence analysis of Newton's method for sections on Riemannian manifolds. Using the notion of a 2-piece L-average Lipschitz condition introduced in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant ?-theory, J. Complexity 24 (2008) 423-451] in combination with the weaker center 2-piece L 1-average Lipschitz condition given by us in this paper, we provide a tighter convergence analysis than the one given in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant ?-theory, J. Complexity 24 (2008) 423-451] which in turn has improved the works in earlier studies such as [R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal. 22 (2002) 359-390; F. Alvarez, J. Bolte and J. Munier, A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math. 8 (2008) 197-226; J. P. Dedieu, P. Priouret and G. Malajovich, Newton's method on Riemannian manifolds: Covariant ?-theory, IMA J. Numer. Anal. 23 (2003) 395-419]. World Scientific Publishing Company.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.