Journal Articles
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Item Modification of the kantorovich-type conditions for newton's method involving twice frechet differentiable operators(2013) Argyros, I.K.; George, S.We expand the applicability of Newton's method for approximating a locally unique solution of a nonlinear equation in a Banach space setting. The nonlinear operator involved is twice Fréchet differentiable. We introduce more precise majorizing sequences than in earlier studied (see [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal. 11 (2004) 103-119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math. 79 (1997) 131-145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217]). This way, our convergence criteria can be weaker; the error estimates tighter and the information on the location of the solution more precise. Numerical examples are presented to show that our results apply in cases not covered before such as [Concerning the convergence and application of Newton's method under hypotheses on the first and second Fréchet derivative, Comm. Appl. Nonlinear Anal. 11 (2004) 103-119; A new semilocal convergence theorem for Newton's method, J. Comp. Appl. Math. 79 (1997) 131-145; A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211-217]. © 2013 World Scientific Publishing Company.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 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 Fréchet-derivative of the operator involved is p-Hölder continuous (p ?(0, 1]). Numerical examples involving two boundary value problems are also provided. © 2014 Korean Mathematical Society.Item Expanding the convergence domain for chun-stanica-neta family of third order methods in banach spaces(Korean Mathematical Society kms@kms.or.kr, 2015) Argyros, I.K.; George, S.; Magreñán Ruiz, Á.A.We present a semilocal convergence analysis of a third order method for approximating a locally unique solution of an equation in a Banach space setting. Recently, this method was studied by Chun, Stanica and Neta. These authors extended earlier results by Kou, Li and others. Our convergence analysis extends the applicability of these methods under less computational cost and weaker convergence criteria. Numerical examples are also presented to show that the earlier results cannot apply to solve these equations. © 2015 Korean Mathematical Society.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 An improved semi-local convergence analysis for a three point method of order 1.839 in banach space(International Publications internationalpubls@yahoo.com, 2015) Argyros, I.K.; Padikkal, P.; George, S.We present a new semi-local convergence analysis for a three point method of order 1.839 in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The advantages of the new approach over earlier ones such as [18] are: weaker and easier to verify convergence conditions. Moreover the radius of convergence is given in an explicit form. Furthermore, uniqueness results are also presented for the first time as far as we know in this paper. Finally, numerical example illustrating the theoretical results is given.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 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 Local convergence for some high convergence order Newton-like methods with frozen derivatives(Springer Nature, 2015) Argyros, I.K.; George, S.We present a local convergence analysis of some families of Newton-like methods with frozen derivatives in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Amat et al. (Appl Math Lett. 25:2209–2217, 2012), Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam, 2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Xiao and Yin (Appl Math Comput, 2015) the local convergence was proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these methods. In this paper we expand the applicability of these methods using only hypotheses on the first derivative and Lipschitz constants. Numerical examples are also presented in this study. © 2015, Sociedad Española de Matemática Aplicada.