Faculty Publications
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Publications by NITK Faculty
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Item On extended convergence domains for the newton-kantorovich method(Publishing House of the Romanian Academy Calea 13 Septembrie nr. 13, Sector 5, 050711. P.O. Box 5-42, Bucuresti, 2014) Argyros, I.K.; George, S.We present results on extended convergence domains and their applications for the Newton-Kantorovich method (NKM), using the same information as in previous papers. Numerical examples are provided to emphasize that our results can be applied to solve nonlinear equations using (NKM), in contrast with earlier results which are not applicable in these cases. © 2014, Publishing House of the Romanian Academy. All rights reserved.Item EXPANDING THE APPLICABILITY OF THE GAUSS-NEWTON METHOD FOR A CERTAIN CLASS OF SYSTEMS OF EQUATIONS(Publishing House of the Romanian Academy, 2016) Argyros, I.K.; George, S.We present a new semi-local convergence analysis of the Gauss-Newton method in order to solve a certain class of systems of equations under a majorant condition. Using a center majorant function as well as a majorant function and under the same computational cost as in earlier studies such as [11]-[13], we present a semilocal convergence analysis with the following advan-tages: weaker sufficient convergence conditions; tighter error estimates on the distances involved and an at least as precise information on the location of the solution. Special cases and applications complete this study. © 2016, Publishing House of the Romanian Academy. All rights reserved.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 Ball Convergence for two-parameter chebyshev-halley-like methods in banach space using hypotheses only on the first derivative(International Publications internationalpubls@yahoo.com, 2017) Argyros, I.K.; George, S.; Verma, R.U.We present a local convergence analysis of a sixth-order method for approximate a locally unique solution of an equation in the Banach space setting. The convergence of this methods is shown in Narang et al. (2016) under hypotheses up to the fourth Fréchet-derivative and the Lipschitz continuity of the third derivative, although only the first derivative appears in the method. In this study we expand the applicability of this method using only hypotheses on the first derivative of the function. Numerical examples are also presented in this study.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 Expanding the Applicability of the Kantorovich’s Theorem for Solving Generalized Equations Using Newton’s Method(Springer, 2017) Argyros, I.K.; George, S.In this paper we consider the Kantorovich’s theorem for solving generalized equations F(x) + Q(x) ? 0 using Newton’s method, where F is a Fréchet differentiable function and Q is a set-valued and maximal monotone function acting between Hilbert spaces. We used our new idea of restricted convergence domains to obtain better location about where the iterates are located leading to a tighter convergence analysis than in the earlier studies and under the same or less computational cost of the majorant functions involved. © 2016, Springer India Pvt. Ltd.Item Ball convergence for an eighth order efficient method under weak conditions in Banach spaces(Springer Nature, 2017) Argyros, I.K.; George, S.; Erappa, S.M.We present a local convergence analysis of an eighth order- iterative method in order to approximate a locally unique solution of an equation in Banach space setting. Earlier studies have used hypotheses up to the fourth 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. © 2016, Sociedad Española de Matemática Aplicada.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 Extending the Mesh Independence For Solving Nonlinear Equations Using Restricted Domains(Springer, 2017) Argyros, I.K.; Sheth, S.M.; Younis, R.M.; Magreñán Ruiz, Á.A.; George, S.The mesh independence principle states that, if Newton’s method is used to solve an equation on Banach spaces as well as finite dimensional discretizations of that equation, then the behaviour of the discretized process is essentially the same as that of the initial method. This principle was inagurated in Allgower et al. (SIAM J Numer Anal 23(1):160–169, 1986). Using our new Newton–Kantorovich-like theorem and under the same information we show how to extend the applicability of this principle in cases not possible before. The results can be used to provide more efficient programming methods. © 2017, Springer (India) Private Ltd.Item Local convergence of bilinear operator free methods under weak conditions(Drustvo Matematicara Srbije drustvomatematicara@yahoo.com, 2018) Argyros, I.K.; George, S.We study third-order Newton-type methods free of bilinear operators for solving nonlinear equations in Banach spaces. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot apply to solve equations but our results can apply are also given in this study. © 2018, Drustvo Matematicara Srbije. All rights reserved.