Local Convergence of an Efficient High Convergence Order Method Using Hypothesis Only on the First Derivative

• Main Authors: Ioannis K. Argyros, Ramandeep Behl, S.S. Motsa
• Format: Article
• Language: English
• Published: MDPI AG 2015-11-01
• Series: Algorithms
• Subjects: Newton-like method; local convergence; efficiency index; optimum method
• Online Access: http://www.mdpi.com/1999-4893/8/4/1076

We present a local convergence analysis of an eighth order three step methodin order to approximate a locally unique solution of nonlinear equation in a Banach spacesetting. In an earlier study by Sharma and Arora (2015), the order of convergence wasshown using Taylor series expansions and hypotheses up to the fourth order derivative oreven higher of the function involved which restrict the applicability of the proposed scheme. However, only first order derivative appears in the proposed scheme. In order to overcomethis problem, we proposed the hypotheses up to only the first order derivative. In this way,we not only expand the applicability of the methods but also propose convergence domain. Finally, where earlier studies cannot be applied, a variety of concrete numerical examplesare proposed to obtain the solutions of nonlinear equations. Our study does not exhibit thistype of problem/restriction.