1. Faculty Publications
Permanent URI for this communityhttps://idr.nitk.ac.in/handle/1/5
Browse
75 results
Search Results
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 Projection method for newton-tikhonov regularization for non-linear ill-posed hammerstein type operator equations(2013) Shobha, M.E.; George, S.An iteratively regularized projection scheme for the ill-posed Hammerstein type operator equation KF(x) = f has been considered. Here F : D(F)X X is a non-linear operator, K : X ? Y is a bounded linear operator, X and Y are Hilbert spaces. The method is a combination of dis- cretized Tikhonov regularization and modified Newton's method. It is assumed that the F?echet derivative of F at x0 is invertible i.e., the ill-posedness of the operator KF is due to the ill-posedness of the linear operator K. Here x0 is an initial approximation to the solution x of KF(x) = f. Adaptive choice of the parameter suggested by Perverzev and Schock(2005) is employed in select- ing the regularization parameter ?. A numerical example is given to test the reliability of the method. 2013 Academic Publications, Ltd.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 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 complexity of extending the convergence region for Traub's method(2020) Argyros, I.K.; George, S.The convergence region of Traub's method for solving equations is small in general. This fact limits its applicability. We locate a more precise region containing the Traub iterations leading to at least as tight Lipschitz constants as before. Our convergence analysis is finer, and obtained without additional conditions. The new theoretical results are tested on numerical examples that illustrate their superiority over earlier results. 2019Item On the convergence of Broyden's method with regularity continuous divided differences and restricted convergence domains(2017) Argyros, I.K.; George, S.We present a semilocal convergence analysis for Broyden's method with regularly continuous divided differences in a Banach/Hilbert space setting. By using: center-Lipschitz-type conditions defining restricted convergence domains, at least as weak hypotheses and the same computational cost as in [6] we provide a new convergence analysis for Broyden's method with the following advantages: larger convergence domain; finer error bounds on the distances involved, and at least as precise information on the location of the solution. 2017 Journal of Nonlinear Functional Analysis.Item On the "terra incognita" for the newton-kantrovich method with applications(2014) Argyros, I.K.; Cho, Y.J.; George, S.In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]-[13], in some interesting cases, provided that the Fr chet-derivative of the operator involved is p-H lder continuous (p ?(0, 1]). Numerical examples involving two boundary value problems are also provided. 2014 Korean Mathematical Society.Item On extended convergence domains for the newton-kantorovich method(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 On improving the semilocal convergence of newton-type iterative method for ill-posed Hammerstein type operator equations(2013) Shobha, M.E.; George, S.George and Pareth( 2012), presented a quartically convergent Two Step Newton type method for approximately solving an ill-posed operator equation in the finite dimensional setting of Hilbert spaces. In this paper we use the analogous Two Step Newton type method to approximate a solution of ill-posed Hammerstein type operator equation.