A modified Newton-like method for nonlinear equations
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A modified Newton-like method for nonlinear equations Song Wu1 · Haijun Wang1 Received: 18 November 2019 / Revised: 30 April 2020 / Accepted: 30 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, a modified Newton-like method for nonlinear equations is proposed, which is based on a rational model. In the rational model, the Jacobian matrix is approximated by a matrix Dk which satisfies a new quasi-Newton equation. The new quasi-Newton equation exploits addition information by assuming a quadratic relationship between the information from the last three iterations. Under mild assumptions, the local convergence properties of the new method is presented. Numerical results illustrate that the modified Newton-like method is efficient. Keywords Newton-like method · Nonlinear equations · Quasi-Newton equation · Rational model · Local convergence Mathematics subject classification 65K05 · 90C53
1 Introduction We consider the systems of nonlinear equations F(x) = 0, Rn
(1.1)
where F : → is continuously differentiable and F(x) = ( f 1 (x), f 2 (x), . . . , f n (x))T , T x = (x1 , x2 , . . . , xn ) . The classical iterative method for solving (1.1) is Newton’s method (Broyden 1971; Dennis and Schnabel 1993; Kelley 2003), which generates an iterate sequence xk+1 by Rn
xk+1 = xk − J (xk )−1 F(xk ), k = 0, 1, 2, . . . ,
(1.2)
where J (xk ) is the Jacobian matrix of F(x) at xk . This method converges rapidly from any sufficiently good starting point. It is expensive to compute and store the Jacobian matrix directly at each iteration. In recent years, quasi-Newton method (Broyden 1965; Dennis and
Communicated by Joerg Fliege.
B 1
Haijun Wang [email protected] School of Mathematics, China University of Mining and Technology, Xuzhou 221116, People’s Republic of China 0123456789().: V,-vol
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S. Wu, H. Wang
Mor 1977; Martínez 2000; Al-Baali et al. 2014; Martínez and Ochi 1982) has greatly improved the computational efficiency. The most successful and simplest quasi-Newton method for nonlinear equations is the Broyden’s method (Broyden 1965). The classical Broyden’s method is xk+1 = xk − Dk−1 F(xk ), k = 0, 1, 2, . . . ,
(1.3)
where matrix Dk is an approximation of J (xk ) and satisfies the following equation Dk sk−1 = yk−1 ,
(1.4)
where sk−1 = xk − xk−1 , yk−1 = F(xk ) − F(xk−1 ), and Dk = Dk−1 +
T (yk−1 − Dk−1 sk−1 )sk−1 T s sk−1 k−1
.
(1.5)
Over the years, there are various methods (Frontini and Sormani 2004; Cruz et al. 2006; Grau-Sánchez et al. 2007; Zhang et al. 1999; Zhang and Xu 2001; Wei et al. 2006) for the solution of (1.1). For example, Frontini and Sormani (2004) presented a third-order method from quadrature formulae for solving nonlinear equations, Cruz et al. (2006) investigated a spectral residual method without gradient information for finding solution of nonlinear equations, Grau-Sánchez et al. (2007) discussed accelerated iterative methods for systems of nonlinear equations, and so on. Most of these methods only used t
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