Computational efficiency of some combined methods for polynomial equations q Ivan Petkovic ´ Department of Computer Science, Faculty of Electronic Engineering, University of Niš, 18000 Niš, Serbia article info Keywords: Computational efficiency Polynomial roots Combined methods Simultaneous methods Improvement of convergence abstract The iterative methods for the simultaneous determination of all simple complex zeros of algebraic polynomials, based on the fixed point relation of Ehrlich’s type, are considered. Using the iterative correction appearing in the Jarratt method of the fourth order, it is proved that the convergence rate of the modified Ehrlich method is increased from 3 to 6. This acceleration of the convergence is obtained with few additional numerical opera- tions which means that the proposed combined method possesses very high computational efficiency. Moreover, the convergence rate can be further accelerated using the Gauss–Sie- del approach (single-step or serial mode). A great part of the paper is devoted to the com- putational aspects of the discussed methods, including numerical examples. A comparison procedure shows that the new iterative method is more efficient than existing methods in the considered class. Ó 2008 Elsevier Inc. All rights reserved. 1. Computational efficiency of root-finding methods During the last decade a ‘‘flood” of root finding methods have appeared in various journals for applied and computational mathematics, often without a proper justification. Namely, most of these nonprogressive methods are either rediscovered ones or they do not improve the computational efficiency of the existing methods. In this paper we propose a combined method whose computational efficiency is the highest within the class of iterative methods for the simultaneous approxi- mation of simple complex roots of algebraic polynomials, based on fixed point relations. We have restricted ourselves to this class of methods since they are frequently used in practice, possess a high computational efficiency and they can be success- fully implemented on parallel computers. Many papers devoted to iterative methods for solving nonlinear equations, including algebraic equations, discuss and estimate the efficiency of these methods only on the basis of numerical examples which play, at the same time, the key role in the comparison with other methods. In fact, this approach is deficient since it strongly depends on the tested functions and initial approximations. The estimation of computational efficiency by the amount of computational work, most fre- quently expressed by the central processor unit (CPU) time, is obviously considerably more realistic. A method is more effi- cient as its amount of computational work is smaller. In other words, the most efficient method (within the tested iterative methods) is the one which satisfies the required termination criterion (for example, the accuracy of produced approxima- tions to the zeros attains a given tolerance s) for the smallest CPU time. The efficiency of an iterative method ðMÞ can be successfully estimated using the coefficient of efficiency given by EðMÞ¼ r j=h : ð1Þ 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.08.005 q This work was supported by the Serbian Ministry of Science under Grant 144024. E-mail addresses: ivanp@elfak.ni.ac.yu, ivanpetkovic@gmail.com Applied Mathematics and Computation 204 (2008) 949–956 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc