A cubic convergent iterative method for enclosing simple roots of nonlinear equations q P.K. Parida, D.K. Gupta * Department of Mathematics, Indian Institute of Technology, Kharagpur, India Abstract The aim of this paper is to propose a cubic convergent iterative method for solving the nonlinear equation f(x) = 0, where f : ½a; b R ! R is a continuously differentiable function. The method is the hybrid of a Newton-like method and the bisection method and is slightly modified form of that proposed by Zhu and Wu [A free-derivative iteration method of order three having convergence of both point and interval for nonlinear equations, Applied Mathematics and Computation 137 (2003) 49–55]. It starts with a suitably chosen initial approximation x 0 to the root r of f(x)=0 and generates two sequences, one sequence of successive iterates {x n }, and the other sequence of intervals {[a n , b n ]} con- taining the root r. It is shown that the sequence of diameters {(b n  a n )} and the sequence of iterates {(x n  r)} both con- verge cubically to 0 simultaneously. The convergence analysis is carried out for the method. The method is tested on a number of numerical examples and results obtained show that the proposed method is better in comparison with the method given by Zhu and Wu (2003). Ó 2006 Elsevier Inc. All rights reserved. Keywords: Nonlinear equation; Bisection method; Newton like method; Cubic convergence 1. Introduction One of the most important tasks of scientific computing is that of finding the roots of nonlinear equations and/or systems of nonlinear equations. This occurs in many situations. For example, the locations of the extre- mal points of a function require finding the roots of the derivatives of that function. It is shown in [5] that the nonlinear boundary value problems solved by difference methods yield the systems of nonlinear equations whose solutions are the solutions of the boundary value problems. Areas such as chemical engineering, trans- portation, operation research and many others involve frequently solving nonlinear algebraic equations either 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.071 q This work is supported by financial grant, CSIR (No: 10-2(5)/2004(i)-EU II), India. * Corresponding author. E-mail addresses: pradipparida@yahoo.com (P.K. Parida), dkg@maths.iitkgp.ernet.in (D.K. Gupta). Applied Mathematics and Computation 187 (2007) 1544–1551 www.elsevier.com/locate/amc