A family of higher order derivative free methods for nonlinear systems with local convergence analysis
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A family of higher order derivative free methods for nonlinear systems with local convergence analysis S. Bhalla1 · S. Kumar1 · I. K. Argyros2 · Ramandeep Behl3
Received: 16 November 2017 / Accepted: 1 June 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract A wide general class of Ostrowski’s families without memory proposed by Behl et al. (Int J Comput Math 90(2):408–422, 2013) is being extended to solve systems of nonlinear equations. This extension uses multidimensional divided differences of first order. Many more new derivative free iterative families with higher order local convergence are presented. In addition, the proposed iterative family for α1 = R − {0} and α2 = 0 are special cases of Grau et al. (J Comput Appl Math 237:363–372, 2013) for iterative schemes of fourth and sixth orders. The computational efficiency is compared with some known methods. It is proved that the proposed methods are equally competent with their existing counter parts. Moreover, we present the local convergence analysis of the proposed family of methods based on Lipschitz constants and hypotheses on the divided difference of order one in the more general settings of a Banach space. We expand this way the applicability of these methods, since we used higher derivatives to show convergence of the method in Sect. 3 although such derivatives do not appear in these methods. Numerical experiments are performed which support the theoretical results. Keywords System of nonlinear equations · Order of convergence · Steffensen’s method · Computational efficiency · Derivative free methods Mathematics Subject Classification MSC 65H10 · 41A58 · 65Y20
Communicated by Andreas Fischer.
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Ramandeep Behl [email protected]
1
School of Mathematics, Thapar Institute of Engineering and Technology, Patiala, Punjab 147004, India
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Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA
3
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
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1 Introduction One of the most basic and earliest problems of numerical analysis concerns with finding efficiently and accurately the approximate solution of a nonlinear system G(X ) = 0, where G(X ) = ( g1 (X ), g2 (X ), . . . , gt (X ))T , X = (x1 , x2 , . . . , xt )T and G : Rt → Rt is a sufficiently differentiable vector function. Analytical methods for solving such problems are non-existent, and therefore, it is only possible to obtain approximate solutions, by relying on numerical techniques based on iteration procedures. The most simple and common iterative method for this purpose is the Newton’s method (Kelley 2003; Traub 1964), which converges quadratically and is defined by −1 X k+1 = X k − G (X k ) G(X k ), k = 0, 1, 2, . . . , −1 is the inverse of first Fréchet derivative G (X k ) of the function of G(X ). where G (X k ) The practice of Numerical Functional Analysis for approximating solutions iteratively is essentially connected to Newton-like metho
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