Optimal Newton-Secant like methods without memory for solving nonlinear equations Mehdi Salimi a∗ Taher Lotfi b † Somayeh Sharifi b ‡ Stefan Siegmund a§ a Department of Mathematics, Technische Universit¨at Dresden, 01062 Dresden, Germany b Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran Abstract We construct two optimal Newton-Secant like iterative methods for solving non-linear equa- tions. The proposed classes have convergence order four and eight and cost only three and four function evaluations per iteration, respectively. These methods support the Kung and Traub conjecture and possess a high computational efficiency. We conclude with numerical experiments and a comparison with some existing optimal methods. Keywords: Multi-point iterative methods; Newton-Secant method; Kung and Traub’s con- jecture. 1 Introduction A main tool for solving nonlinear problems is the approximation of simple roots x ∗ of a nonlinear equation f (x ∗ ) = 0 with a scalar function f : D ⊂ R → R which is defined on an open interval D (see e.g. [18, 23] and the references therein). The secant method is a simple root-finding algorithm which can be traced back to a historic precursor called “rule of double false position” [19]. A modern way to view the secant method would be to replace the derivative in the Newton-Raphson method x n+1 = x n − f (xn) f ′ (xn) by a finite-difference approximation. The Newton-Raphson method is one of the most widely used algorithms for finding roots. It is of second order and requires two evaluations for each iteration step, one evaluation of f and one of f ′ . Newton-Raphson iteration is an example of a one-point iteration, i.e. in each iteration step the evaluations are taken at one point. Multiple-point methods evaluate at several points in each iteration step and in principle allow for a higher convergence order with a lower number of function evaluations. Kung and Traub [14] conjectured that no multi-point method without memory with k evaluations could have a convergence order larger than 2 k−1 . A multi-point method with convergence order 2 k−1 is called optimal. In this paper we construct two new optimal multi-point methods. We present a two-point iteration with convergence order four which requires two evaluations of f and one evaluation of f ′ and a * Corresponding author: mehdi.salimi@mailbox.tu-dresden.de † lotfi@iauh.ac.ir ‡ somayeh.sharifi69@yahoo.com § stefan.siegmund@tu-dresden.de 1