An improved three-term derivative-free method for solving nonlinear equations
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An improved three-term derivative-free method for solving nonlinear equations Auwal Bala Abubakar1,3 · Poom Kumam1,2 Received: 28 April 2018 / Revised: 28 July 2018 / Accepted: 29 August 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract In this work, we proposed an improved three-term derivative-free method for solving system of nonlinear equations, which combines the ideas in Yan et al. (J Comput Appl Math 234:649–657, 2010) and Yuan and Zhang (J Comput Appl Math 286:186–195, 2015) with the projection method. The global convergence of the proposed method was established under some conditions. Numerical experiment shows that the method is efficient and promising compared to some existing ones. Keywords Non-linear equations · Monotone equations · Derivative-free method · Projection method Mathematics Subject Classification 5K05 · 90C06 · 90C52 · 90C56
1 Introduction We consider a nonlinear equation of the form F(x) = 0,
(1)
Communicated by Andreas Fischer.
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Poom Kumam [email protected]; [email protected]
1
KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
2
KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCs), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand
3
Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano, Kano, Nigeria
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A. B. Abubakar, P. Kumam
where F : Rn → Rn is continuous and monotone, i.e., (F(x) − F(y))T (x − y) ≥ 0, ∀x, y ∈ Rn .
(2)
There are so many applications of such problems such as ballistic trajectory computation and vibration systems (Wang et al. 2015; Zeidler 2013), the power flow of equations (Chen et al. 2012; Chuanwei and Yiju 2009), economic equilibrium problem (Dirkse and Ferris 1995; Wang et al. 2014), etc. It is a known fact that the solution set of a monotone equation is convex except if it is empty. Also, a nonlinear complementarity problem (NCP) is a problem of finding x ∈ Rn such that x ≥ 0,
F(x) ≥ 0, x T F(x) ≥ 0
(3)
Complementarity problems are related to mathematical programming, variational inequalities, fixed point problems and mixed strategy problems (see Chen and Li 2001; Facchinei and Pang 2007; Han et al. 2006; Yong et al. 2014; Yu et al. 2006; Chen et al. 1998; Zhu et al. 2007). For example, in Haitao et al. (2012) proposed a new class of smoothing functions for nonlinear complementarity problem which contains the well-known Chen–Harker–Kanzow– Smale smoothing function and Huang–Han–Chen smoothing function as special cases, and then present a smoothing inexact Newton algorithm for the P0 nonlinear complementarity problem. The global and local superlinear
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