Appl. Math. Inf. Sci. 8, No. 5, 2189-2193 (2014) 2189 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080512 Derivative-Free Iterative Methods for Solving Nonlinear Equations Farooq Ahmed Shah 1,∗ , Muhammad Aslam Noor 2 and Moneeza Batool 2 1 Department of Mathematics, COMSATS Institute of Information Technology, Attock Campus, Pakistan 2 Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan Received: 15 Aug. 2013, Revised: 12 Nov. 2013, Accepted: 13 Nov. 2013 Published online: 1 Sep. 2014 Abstract: In this paper, we suggest and analyze some new derivative free iterative methods for solving nonlinear equation f (x)= 0 using a suitable transformation. We also give several examples to illustrate the efficiency of these methods. Comparison with other similar method is also given. These new methods can be considered as alternative to the developed methods. This technique can be used to suggest a wide class of new iterative methods for solving nonlinear equations. Keywords: Nonlinear equation, Convergence, Steffensen’s method, derivative-free method, Examples. 1 Introduction One of the most frequently occurring problems in scientific work is to locate the approximate solution of a nonlinear equation f (x)= 0 (1) Analytical methods for solving such equations are almost nonexistent and therefore, it is only possible to obtain approximate solutions by relying on numerical techniques based on iteration procedures [1, 2, 3, 4, 5, 6, 8, 9]. If the function is not known explicitly or the derivative is difficult to compute, a method that uses only computed values of the function is more appropriate. Some of the more classical numerical methods for solving nonlinear equations without using derivative [9] include the bisection method, secent method and regula falsi method. These are the basic methods but have slow convergence toward the solution. Newton’s method, which is simple and converges quadratically [9], is probably the best known and most widely used algorithm which includes the derivative of the function. However, Steffensen’s method [3, 9] x n+1 = x n − [ f (x n )] 2 f (x n + f (x n )) − f (x n ) n = 0, 1, 2, 3, ... is variation of Newton’s method which does not employ the derivative of the function. In this method the derivative is approximated by the forward difference scheme. Steffensen’s method has same order of convergence as Newton’s method. Based on the approximation of the first derivative, we construct some derivative-free iterative methods for solving nonlinear equations. 2 Iterative methods In this section, we construct some iterative methods for solving nonlinear equations. We use approximation of first derivative of the function to obtain derivative-free methods. Let us approximate the first derivative of the function f (x n )= 0, at the current iteration x n by f ′ (x n ) ≈ g(x n )= f (x n + bf (x n )) − f (x n ) bf (x n ) , (2) ∗ Corresponding author e-mail: farooqhamdani@gmail.com c  2014 NSP Natural Sciences Publishing Cor.