Article

A composite third order Newton–Steffensen method for solving nonlinear equations

Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, 148 106 District Sangrur, India
Applied Mathematics and Computation (Impact Factor: 1.35). 01/2005; DOI: 10.1016/j.amc.2004.10.040
Source: DBLP

ABSTRACT In this paper, we suggest a third-order method formed by the composition of Newton and Steffensen methods for finding simple and real roots of a nonlinear equation in single variable. Per iteration the formula requires two evaluations of the function and single evaluation of the derivative. Experiments show that the method is suitable in the cases where Newton and Steffensen methods fail.

2 Bookmarks
 · 
250 Views
  • [Show abstract] [Hide abstract]
    ABSTRACT: We study the solving of nonlinear equations by an iterative method of Aitken type, which has the interpolation nodes controlled by the Newton method. We obtain a local convergence result which shows that the q-convergence order of this method is 6 and its efficiency index is $\sqrt[5]{6},$ which is higher than the efficiency index of the Aitken or Newton methods. Monotone sequences are obtained for initial approximations farther from the solution, if they satisfy the Fourier condition and the nonlinear mapping satisfies monotony and convexity assumptions on the domain.
    Numerical Algorithms 02/2013; 62(2). · 1.13 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: We study the local convergence of some Aitken–Steffensen–Hermite type methods of order three. We obtain that under some reasonable conditions on the monotony and convexity of the nonlinear function, the iterations offer bilateral approximations for the solution, which can be efficiently used as a posteriori estimations.
    Applied Mathematics and Computation 02/2011; 217(12):5838–5846. · 1.35 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: A special class of neural dynamics called Zhang dynamics (ZD), which is different from gradient dynamics (GD), has recently been proposed, generalized, and investigated for solving time-varying problems by following Zhang et al.’s design method. In view of potential digital hardware implemetation, discrete-time ZD (DTZD) models are proposed and investigated in this paper for solving nonlinear time-varying equations in the form of $f(x,t)=0$ . For comparative purposes, the discrete-time GD (DTGD) model and Newton iteration (NI) are also presented for solving such nonlinear time-varying equations. Numerical examples and results demonstrate the efficacy and superiority of the proposed DTZD models for solving nonlinear time-varying equations, as compared with the DTGD model and NI.
    Numerical Algorithms 12/2013; · 1.13 Impact Factor

Full-text (2 Sources)

Download
7 Downloads
Available from