NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl. 2009; 16:971–994 Published online 3 August 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nla.661 Chebyshev-like root-finding methods with accelerated convergence M. S. Petkovi´ c 1, ∗, † , L. Ranˇ ci´ c 1 , L. D. Petkovi´ c 2 and S. Ili´ c 3 1 Faculty of Electronic Engineering, Department of Mathematics, University of Niˇ s, 18000 Niˇ s, Serbia 2 Faculty of Mechanical Engineering, Department of Mathematics, University of Niˇ s, 18000 Niˇ s, Serbia 3 Faculty of Science, Department of Mathematics, University of Niˇ s, 18000 Niˇ s, Serbia SUMMARY Iterative methods for the simultaneous determination of simple or multiple complex zeros of a polynomial, based on a cubically convergent Chebyshev method, are considered. Using Newton’s and Halley’s correc- tions the convergence of the basic method of the fourth order is increased to five and six, respectively. The improved convergence is achieved with negligible number of additional calculations, which significantly increases the computational efficiency of the accelerated methods. One of the most important problems in solving polynomial equations, the construction of initial conditions that enable both guaranteed and fast convergence, is also studied for the proposed methods. These conditions are computationally verifiable since they depend only on initial approximations, the polynomial coefficients and the polynomial degree, which is of practical importance. Finally, modified methods of Chebyshev’s type for finding multiple zeros and single-step methods based on the Gauss–Seidel approach are constructed. Copyright 2009 John Wiley & Sons, Ltd. Received 6 July 2008; Revised 13 March 2009; Accepted 26 April 2009 KEY WORDS: polynomial zeros; Chebyshev’s method; simultaneous methods; acceleration of convergence; initial conditions Dedicated to Professor Ivo Marek on the occasion of his 75th Birthday 1. INTRODUCTION Iterative methods for the simultaneous approximation of zeros of algebraic polynomials are frequently used powerful tool in numerical analysis as well as various technical disciplines, see, ∗ Correspondence to: M. S. Petkovi´ c, Faculty of Electronic Engineering, Department of Mathematics, University of Niˇ s, 18000 Niˇ s, Serbia. † E-mail: miodragpetkovic@gmail.com Contract/grant sponsor: Serbian Ministry of Science; contract/grant number: 144024 Copyright 2009 John Wiley & Sons, Ltd.