Ninth-order Three-point Derivative-Free Family of Iterative Methods for Nonlinear Equations Malik Zaka Ullah a , A.S. Al-Fhaid a , Fayyaz Ahmad b,1,∗ a Department of Mathematics, King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia b Dept. de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Barcelona 08036, Spain Abstract In this note, we present ninth-order derivative-free family of iterative methods for nonlinear equations. The weight functions are introduced in the iterative scheme which actually constitute a family of iterative methods. The right selection of weight function provides a rapid convergence towards root of the nonlinear equation. The Steffensen’s method is used along with a free parameter κ 1 as a first step in the construction for the underlaying iterative family. The proposed iterative scheme requires five function evaluations which is not optimal in the sense of the Kung and Traub conjecture [5] but the weight functions are very simple and require few computations. In the last, numerical experiments are performed to show the numerical performance of different derivative-free iterative methods. The computational order of convergence for the iterative family is also calculated to confirm numerically the claimed order of convergence. Keywords: Non-linear equations, Steffensen’s method, Derivative-free, Iterative methods 1. Introduction Let f be a sufficiently smooth function of single variable in some neighborhood D of α, where α satisfies f ? (α) ? 0 and f (α) = 0. The well-known Newton method is defined as x n+1 = x n − f ( x n ) f ? ( x n ) , (1) which is quadratically convergent. The Steffensen’s approximation for the first order derivative: f ( x n − κ 1 f ( x n )) ≈ f ( x n ) − κ 1 f ( x n ) f ? ( x n ), κ 1 f ( x n ) f ? ( x n ) ≈ f ( x n ) − f ( x n − κ 1 f ( x n )), f ? ( x n ) ≈ 1 κ 1 f ( x n ) − f ( x n − κ 1 f ( x n )) f ( x n ) . (2) After the substitute of the derivative approximation (2) in (1), we get Steffensen’s second order accurate derivative-free iterative method for non-linear equations [1]. w n = x n − κ 1 f ( x n ), x n+1 = x n − κ 1 f ( x n ) 2 f ( x n ) − f (w n ) . (3) ∗ Corresponding author Email addresses: mzhussain@kau.edu.sa ( Malik Zaka Ullah ), aalfhaid@hotmail.com,aalfhaid@kau.edu.sa (A.S. Al-Fhaid), fayyaz.ahmad@upc.edu ( Fayyaz Ahmad ) 1 This research was supported by Spanish MICINN grants AYA2010-15685. April 8, 2013