Increasing the order of convergence for iterative methods in Banach space under weak conditions

Ioannis K. Argyros; Santhosh George

doi:10.26637/MJM0602/0016

Abstract

We study the method considered in Xiao and Yin (2015), for solving systems of nonlinear equations, modified suitably to include the nonlinear equations in Banach spaces. The novelty of this study lies in the fact that our conditions are weaker than the conditions used in earlier studies. This way we extend the applicability of the method. Numerical examples are also given in this study where earlier results cannot apply to solve equations but our results can apply.

Keywords:

Newton-type method, radius of convergence, local convergence, restricted convergence domains

Mathematics Subject Classification:

Mathematics

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Ioannis K. Argyros Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA.
Santhosh George Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka-575 025, India. https://orcid.org/0000-0002-3530-5539

Pages: 396-401
Date Published: 01-04-2018
Vol. 6 No. 02 (2018): Malaya Journal of Matematik (MJM)

S. Amat, S. Busquier, Advances in iterative methods for nonlinear equations, SEMS SIMAI springer series, Switzerlands, 2016.

I.K. Argyros, Computational theory of iterative methods. Series: Studies in Computational Mathematics, 15, Editors: C.K.Chui and L. Wuytack, Elsevier Publ. Co. New York, U.S.A, 2007.

I.K.Argyros and F.Szidarovszky, The theory and Applications of Iterative methods,CRC Press,Boca Raton Florida,USA, 1993.

I.K. Argyros and H. Ren, Improved local analysis for certain class of iterative methods with cubic convergence, Numerical Algorithms, 59(2012), 505-521.

I. K. Argyros and A. A. Magreñán, Local convergence analysis of proximal Gauss-Newton method for penalized nonlinear least square problems, Appl. Math. Comput.V. $241,15(2014), 401-408$

I.K.Argyros, Yeol Je Cho and S. George, Local convergence for some third-order iterative methods under weak conditions, J. Korean Math. Soc. 53 (2016), No. 4, pp. $781-793$.

I.K.Argyros and S. George, Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions, SSN 0008-0624, Calcolo, DOI 10.1007/s10092-015-0163-y.

I. K. Argyros and A. A. Magreñán, Iterative methods and their dynamics with applications, CRC Press, New York, USA, 2017.

A. Cordero, J. Hueso, E. Martinez, J. R. Torregrosa, A modified Newton-Jarratt's composition, Numer. Algor. $55,(2010), 87-99$.

A. Cordero, J. R. Torregrosa, Variants of Newton's method for functions of several variables, Appl.Math. Comput. 183,(2006), 199-208.

A. Cordero, J. R. Torregrosa, Variants of Newton's method using fifth order quadrature formulas, Appl.Math. Comput. 190,(2007), 686-698.

G.M Grau-Sanchez, A.Grau, M .Noguera, On the computational efficiency index and some iterative methods for solving systems of non-linear equations, J. Comput. Appl Math., 236 (2011), 1259-1266.

J. M. Gutiérrez and M. A. Hernández, Newton's method under weak Kantorovich conditions, IMA J. Numer. Anal., 20 (2000), 521-532.

M. A. Noor and M. Waseem, Some iterative methods for solving a system of nonlinear equations, Computers and Mathematics with Applications, 57, (2009), 101-106.

W.C. Rheinboldt,An adaptive continuation process for solving systems of nonlinear equations, In: Mathematical models and numerical methods (A.N.Tikhonov et al. eds.) pub.3, (1977), 129-142 Banach Center, Warsaw Poland.

J.R. Sharma, P.K. Gupta, An efficient fifth order method for solving systems of nonlinear equations, Comput. Math. Appl. 67, (2014), 591-601.

F. A. Shah and M. A. Noor, Some numerical methods for solving nonlinear equations by using decomposition technique, Appl. Math. Comput. 251(2015), 378-386.

J.F.Traub, Iterative methods for the solution of equations, AMS Chelsea Publishing, 1982.

X. Y. Xiao and H. W. Yin, Increasing the order of convergence for iterative methods to solve nonlinear equations, Calcolo, (2015), DOI 10.1007/ s10092-015-0149-9.