Modified Jarratt method for computing multiple roots Janak Raj Sharma a, * , Rajni Sharma b a Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148 106, Sangrur, India b Department of Applied Sciences, D.A.V. Institute of Engineering and Technology, Kabirnagar, Jalandhar, India article info Keywords: Rootfinding Jarratt method Multiple roots Order of convergence Efficiency abstract In this paper, we present a fourth order method for computing multiple roots of nonlinear equations. The method is based on Jarratt scheme for simple roots [P. Jarratt, Some efficient fourth order multipoint methods for solving equations, BIT 9 (1969) 119–124]. The method is optimal, since it requires three evaluations per step, namely one evaluation of function and two evaluations of first derivative. The efficacy is tested on a number of relevant numerical problems. It is observed that the present scheme is competitive with other sim- ilar robust methods. Ó 2010 Elsevier Inc. All rights reserved. 1. Introduction Construction of the iterative methods of optimal order for multiple roots is one of the difficult problems in numerical analysis. For information on the optimal order, see for example [1,2]. In this paper, we consider the problem of finding multi- ple roots and construct an iterative method of optimal order four. Traub [3] has studied in detail the multipoint iterative methods, that is, the methods which calculate new approximations to a root by sampling f(x), and its derivatives for a number of values of the independent variable, at each step. However, this study was restricted to simple roots only. In particular, Jarratt [4] considered the following multipoint iterative scheme for simple roots x nþ1 ¼ x n  a 1 w 1 ðx n Þ a 2 w 2 ðx n Þ a 3 w 2 2 ðx n Þ w 1 ðx n Þ ; ð1Þ where w 1 ðx n Þ¼ f ðx n Þ f 0 ðx n Þ ; w 2 ðx n Þ¼ f ðx n Þ f 0 ðy n Þ ; y n ¼ x n þ bw 1 ðx n Þ and obtained a fourth order method for the parameters a 1 ¼ 5 8 ; a 2 ¼ 0; a 3 ¼ 3 8 , and b ¼ 2 3 . It can be seen easily that the Jarr- att’s method has optimal fourth order convergence. In the case of multiple roots, however, this scheme with same parametric values loses fourth order convergence. In fact, the error for this case can be given as e nþ1 ¼ 1  5 8m  3 8m 1  2 3m   22m  ! e n þ Oe 2 n  ; where e n = x n  a and m is the multiplicity of the root a. That means the convergence is linear for multiple roots. 0096-3003/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.06.031 * Corresponding author. E-mail addresses: jrshira@yahoo.co.in (J.R. Sharma), rajni_gandher@yahoo.co.in (R. Sharma). Applied Mathematics and Computation 217 (2010) 878–881 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc