Journal Articles
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Item Extending the applicability of Newton’s method using Wang’s– Smale’s ?–theory(North University of Baia Mare Office_CJEES@yahoo.ro 76 Victoriei Baia Mare 430 122, 2017) Argyros, I.K.; George, S.We improve semilocal convergence results for Newton’s method by defining a more precise domain where the Newton iterate lies than in earlier studies using the Smale’s ?– theory. These improvements are obtained under the same computational cost. Numerical examples are also presented in this study to show that the earlier results cannot apply but the new results can apply to solve equations. © 2017, North University of Baia Mare. All rights reserved.Item Expanding the applicability of the Gauss-Newton method for convex optimization under restricted convergence domains and majorant conditions(Kyungnam University Press jongkyuk@kyungnam.ac.kr, 2017) Argyros, I.K.; George, S.Using our new idea of restricted convergent domains, new semi-local convergence analysis of the Gauss-Newton method for solving convex composite optimization problems is presented. Our convergence analysis is based on a combination of a center-majorant and majorant function. The results extend the applicability of the Gauss-Newton method under the same computational cost as in earlier studies using a majorant function or Wang's condition or Lipchitz condition. The special cases and applications include regular starting points, Robinson's conditions, Smale's or Wang's theory. © 2017 Kyungnam University Press.Item On the convergence of Newton-like methods using restricted domains(Springer New York LLC barbara.b.bertram@gsk.com, 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 Convergence of Stirling’s Method for Fixed Points Under Not Necessarily Contractive Hypotheses(Springer, 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 Inexact Newton’s Method to Nonlinear Functions with Values in a Cone Using Restricted Convergence Domains(Springer, 2017) Argyros, I.K.; George, S.; Erappa, S.M.Using our new idea of restricted convergence domains, a robust convergence theorem for inexact Newton’s method is presented to find a solution of nonlinear inclusion problems in Banach space. Using this technique, we obtain tighter majorizing functions. Consequently, we get a larger convergence domain and tighter error bounds on the distances involved. Moreover, we obtain an at least as precise information on the location of the solution than in earlier studies. Furthermore, a numerical example is presented to show that our results apply to solve problems in cases earlier studies cannot. © 2017, Springer (India) Private Ltd.Item Unified semi-local convergence for k-Step iterative methods with flexible and frozen linear operator(MDPI AG indexing@mdpi.com Postfach Basel CH-4005, 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 Extended convergence analysis of the newton-hermitian and skew-Hermitian splitting method(MDPI AG indexing@mdpi.com Postfach Basel CH-4005, 2019) Argyros, I.K.; George, S.; Godavarma, G.; Magreñán Ruiz, A.A.Many problems in diverse disciplines such as applied mathematics, mathematical biology, chemistry, economics, and engineering, to mention a few, reduce to solving a nonlinear equation or a system of nonlinear equations. Then various iterative methods are considered to generate a sequence of approximations converging to a solution of such problems. The goal of this article is two-fold: On the one hand, we present a correct convergence criterion for Newton-Hermitian splitting (NHSS) method under the Kantorovich theory, since the criterion given in Numer. Linear Algebra Appl., 2011, 18, 299-315 is not correct. Indeed, the radius of convergence cannot be defined under the given criterion, since the discriminant of the quadratic polynomial from which this radius is derived is negative (See Remark 1 and the conclusions of the present article for more details). On the other hand, we have extended the corrected convergence criterion using our idea of recurrent functions. Numerical examples involving convection-diffusion equations further validate the theoretical results. © 2019 by the authors.Item On a Two-Step Kurchatov-Type Method in Banach Space(Birkhauser Verlag AG, 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 A Broyden-type Banach to Hilbert space scheme for solving equations(International Publications internationalpubls@yahoo.com, 2019) Argyros, I.K.; George, S.We present a new semi-local convergence analysis for an inverse free Broyden-type Banach to Hilbert space scheme (BTS) in order to approximate a locally unique solution of an equation. The analysis is based on a center-Lipschitz-type condition and our idea of the restricted convergence region. The operators involved have regularly continuous divided differences. This way we provide, weaker sufficient semi-local convergence conditions, tighter error bounds, and a more precise information on the location of the solution. Hence, our approach extends the applicability of BTS under the same hypotheses as before. © 2019, International Publications. All rights reserved.Item Extended semi-local convergence of Newton’s method on lie groups using restricted regions(International Publications internationalpubls@yahoo.com, 2019) Argyros, I.K.; George, S.We extend the applicability of Newton’s method used to approximate a solution of a mapping involving Lie valued operators. Using our idea of the restricted convergence region, we locate a more precise set containing the Newton iterates leading to tighter majorizing sequences than before. This way and under the same computational cost as before, we show the semi-local convergence of Newton’s method with the following advantages over earlier works: weaker sufficient convergence criteria, tighter error bounds on the distances involved and at least as precise information on the location of the solution. © 2019, International Publications. All rights reserved.