Ball convergence for a two-step fourth order derivative-free method for nonlinear equations

Ioannis K. Argyros ¹   ; Ramandeep Behl ²   ; S. S. Motsa ³

¹ Department of Mathematics Sciences Cameron University Lawton, OK 73505, U.S.A.
² Department of Mathematics King Abdulaziz University Jeddah 21589, Saudi Arabia
³ School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Private Bag X01 Scottsville 3209, Pietermaritzburg, South Africa and Department of Mathematics University of Eswatini Private Bag 4 Kwaluseni, Eswatini (Swaziland)

Applicationes Mathematicae, Tome 46 (2019) no. 2, pp. 253-263

Résumé

We present a local convergence analysis of a two-step fourth order derivative-free method in order to approximate a locally unique solution of a nonlinear equation in a real or complex space setting. In an earlier study of Peng et al. (2011), the order of convergence of the method was shown using Taylor series expansions and hypotheses on up to the fourth order derivative or even higher of the function involved. However, no derivative appears in the proposed scheme. That restricts the applicability of the scheme. We expand the applicability of the scheme using only hypotheses on the first order derivative of the function involved. We also give computable radii of convergence, error bounds based on Lipschitz constants, and the range of initial guesses that guarantees convergence of the methods. Numerical examples where earlier studies do not apply but our results do are also given.

Détail

DOI : 10.4064/am2331-7-2017

Keywords: present local convergence analysis two step fourth order derivative free method order approximate locally unique solution nonlinear equation real complex space setting earlier study peng order convergence method shown using taylor series expansions hypotheses fourth order derivative even higher function involved however derivative appears proposed scheme restricts applicability scheme expand applicability scheme using only hypotheses first order derivative function involved computable radii convergence error bounds based lipschitz constants range initial guesses guarantees convergence methods numerical examples where earlier studies apply results given