Journal Articles
Permanent URI for this collectionhttps://idr.nitk.ac.in/handle/123456789/19884
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Item Ball convergence theorems for unified three step Newton-like methods of high convergence order(Touch Briefings jonathan.mckenna@touchbriefings.com, 2015) Argyros, I.K.; George, S.We present a local convergence analysis for eighth-order variants of Newton's method in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as [7]-[11], [20] using hypotheses up to the seventh derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study. © CSP - Cambridge, UK; I&S - Florida, USA, 2015.Item A unified local convergence for jarratt-type methods in banach space under weak conditions(Chiang Mai University, 2015) Argyros, I.K.; George, S.We present a unified local convergence analysis for Jarratt-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Jarratt;Inverse free Jarratt; super-Halley and other high order methods. The convergence ball and error estimates are given for these methods under the same conditions. Numerical examples are also provided in this study. © 2015 by the Mathematical Association of Thailand. All rights reserved.Item Local convergence of modified Halley-Like methods with less computation of inversion(Institute of Mathematics nsjom@dmi.uns.ac.rs, 2015) Argyros, I.K.; George, S.We present a local convergence analysis of a Modified Halley-Like Method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Frèchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Frèchet-derivative [26]. Numerical examples are also provided in this study. © 2015, Institute of Mathematics. All rights reserved.Item Local convergence of a uniparametric halley-type method in banach space free of second derivative(International Publications internationalpubls@yahoo.com, 2015) Argyros, I.K.; George, S.; Mohapatra, R.N.We present a local convergence analysis of a uniparametric Halley-type method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet-derivative [26]. Numerical examples are also provided in this study.Item Ball convergence for a Newton-steffensen-type third-order method(International Publications internationalpubls@yahoo.com, 2015) Argyros, I.K.; George, S.We present a local convergence analysis for a composite Newton-Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypotheses up to the first derivative in contrast to earlier studies such as [1], [5]-[28] using hypotheses up to the second derivative. This way the applicability of these methods is extended under weaker hypotheses. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.Item Local convergence of deformed Halley method in Banach space under Holder continuity conditions(International Scientific Research Publications editorial-office@tjnsa.com, 2015) Argyros, I.K.; George, S.We present a local convergence analysis for deformed Halley method in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Halley and other high order methods under hypotheses up to the first Fréchet-derivative in contrast to earlier studies using hypotheses up to the second or third Fréchet-derivative. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study. © 2015, International Scientific Research Publications. All rights reserved.Item Ball comparison between two optimal eight-order methods under weak conditions(Springer Nature, 2015) Argyros, I.K.; George, S.We present a local convergence analysis of two families of optimal eighth-order methods in order to approximate a locally unique solution of a nonlinear equation. In earlier studies such as Chun and Lee (Appl Math Comput 223:506–519, 2013), and Chun and Neta (Appl Math Comput 245:86–107, 2014) the convergence order of these methods was given under hypotheses reaching up to the eighth 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 derivative. Moreover, we compare the convergence radii and provide computable error estimates for these methods using Lipschitz constants. © 2015, Sociedad Española de Matemática Aplicada.Item Local convergence for some third-order iterative methods under weak conditions(Korean Mathematical Society kms@kms.or.kr, 2016) Argyros, I.K.; Cho, Y.J.; George, S.The solutions of equations are usually found using iterative methods whose convergence order is determined by Taylor expansions. In particular, the local convergence of the method we study in this paper is shown under hypotheses reaching the third derivative of the operator involved. These hypotheses limit the applicability of the method. In our study we show convergence of the method using only the first derivative. This way we expand the applicability of the method. Numerical examples show the applicability of our results in cases earlier results cannot. © 2016 Korean Mathematical Society.Item Local convergence of inexact Gauss-Newton-like method for least square problems under weak Lipschitz condition(International Publications internationalpubls@yahoo.com, 2016) Argyros, I.K.; George, S.We present a local convergence analysis of inexact Gauss-Newton-like method for solving nonlinear least-squares problems in a Euclidian space setting. The convergence analysis is based on a combination of a weak Lipschitz and a center-weak Lipschitz condition. Our approach has the following advantages and under the same computational cost as earlier studies such as [5, 6, 7, 15]: A large radius of convergence; more precise estimates on the distances involved to obtain a desired error tolerance. Numerical examples are also presented to show these advantages.Item Improved local convergence for Euler–Halley-like methods with a parameter(Springer-Verlag Italia s.r.l. springer@springer.it, 2016) Argyros, I.K.; George, S.We present a local convergence analysis for Euler–Halley-like methods with a parameter in order to approximate a locally unique solution of an equation in a Banach space setting. Using more flexible Lipschitz-type hypotheses than in earlier studies such as Huang and Ma (Numer Algorith 52:419–433, 2009), we obtain a larger radius of convergence as well as more precise error estimates on the distances involved. Numerical examples justify our theoretical results. © 2015, Springer-Verlag Italia.