Japan J. Indust. App!. Math., 15 (1998), 295-315 On Euler-like Methods for the Simultaneous Approximation of Polynomial Zeros Miodrag S. PETKOVICt, Slobodan TRICKOVICtt and Djordje HERCEGttt tFaculty of Electronic Engineering, University of Nis, 18000 Nis, Y ugoslavia ttFaculty of Civil Engineering, University of Nis, 18000 Nis, Y ugoslavia ttt Institute of Mathematics, University of Novi Sad, 21000 Novi Sad, Y ugoslavia Received October 24, 1996 In this paper we consider some iterative methods of higher order for the simultaneous determination of polynomial zeros. The proposed methods are based on Euler's third order method for finding a zero of a given function and involve Weierstrass' correction in the case of simple zeros. We prove that the presented methods have the order of convergence equal to four or more. Based on a fixed -point relation of Euler's type, two inclusion methods are derived. Combining the proposed methods in floating-point arithmetic and complex interval arithmetic, an efficient hybrid method with automatic error bounds is constructed. Computational aspect and the implementation of the presented algorithms on parallel computers are given. Key words: Euler's method, zeros of polynomials, simultaneous methods, interval methods, parallel implementation 1. Introduction A great importance of the problem of determining polynomial zeros in the theory and practice (e.g., in the theory of control systems, stability of systems, nonlinear circuits, analysis of transfer functions, various mathematical models, dif- ferential and difference equations, eigenvalue problems and other disciplines) has led to the development of a great number of numerical methods in this field. These numerical methods, which generally take the form of an iterative procedure, have become practically applicable together with the rapid development of digital com- puters some thirty years ago. However, it is not easy to choose the "best" algorithm for a given polynomial equation. Each algorithm usually possesses its own advan- tages and disadvantages. The selection of a zero-finding routine may depend heav- ily on extramathematical consideration such a speed and memory of the computer equipment and trustworthiness of the results. The aim of this paper is the construction of iterative methods for the simulta- neous determination of polynomial zeros based on Euler's methods (see Euler [13] and Traub [30]). Two basic numerical algorithms of Euler's type and their modifica- tions are derived in Sections 2 and 3. These algorithms possess very fast convergence and can be of importance in practical application, especially when they are imple- mented on parallel computers. Their good convergence behaviour makes them to be