Novi Sad J. Math. Vol. 38, No. 2, 2008, 195-207 . A METHOD FOR OBTAINING THIRD-ORDER ITERATIVE FORMULAS Djordje Herceg 1 , Dragoslav Herceg 2 Abstract. We present a method for constructing new third-order meth- ods for solving nonlinear equations. These methods are modifications of Newton’s method. Also, we obtain some known methods as special cases, for example, Halley’s method, Chebyshev’s method, super-Halley method. Several numerical examples are given to illustrate the performance of the presented methods. AMS Mathematics Subject Classification (2000): 47A63,47A75 Key words and phrases: Nonlinear equations, Newton’s method, Third- order method, Iterative methods 1. Introduction In this paper we consider a family of iterative methods for finding a simple root α of nonlinear equationf (x) = 0. We assume that f satisfies (1) f ∈ C 3 [a, b] , f ′ (x) =0, x ∈ [a, b] , f (a) > 0 >f (b) . Under these assumptions the function f has a unique root α ∈ (a, b). Newton’s method is a well-known iterative method for computing approxi- mation of α by using x k+1 = x k − f (x k ) f ′ (x k ) , k =0, 1,... for some appropriate starting value x 0 . Newton’s method quadratically con- verges in some neighborhood of α if f ′ (α) = 0, [4]. The classical Chebyshev-Halley methods which improve Newton’s method are given by x k+1 = x k − f (x k ) f ′ (x k ) ·  1+ t (x k ) 2 (1 − βt (x k ))  , This paper is a part of the scientific research project no. 144006, supported by the Ministry of Science and Technological Development, Republic of Serbia 1 Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovi´ ca 4, 21000 Novi Sad, Serbia, e-mail: herceg@im.ns.ac.yu 2 Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Trg Dositeja Obradovi´ ca 4, 21000 Novi Sad, Serbia, e-mail: hercegd@im.ns.ac.yu