tailieunhanh - Laguerre-like methods for the simultaneous approximation of polynomial multiple zeros

Two new methods of the fourth order for the simultaneous determination of multiple zeros of a polynomial are proposed. The presented methods are based on the fixed point relation of Laguerre's type and realized in ordinary complex arithmetic as well as circular complex interval arithmetic. The derived iterative formulas are suitable for the construction of modified methods with improved convergence rate with negligible additional operations. | Yugoslav Journal of Operations Research 16 (2006), Number 1, 31-44 LAGUERRE-LIKE METHODS FOR THE SIMULTANEOUS APPROXIMATION OF POLYNOMIAL MULTIPLE ZEROS Miodrag PETKOVIĆ, Lidija RANČIĆ, Dušan MILOŠEVIĆ Faculty of Electronic Engineering, University of Niš Serbia and Montenegro Received: April 2004 / Accepted: May 2005 Abstract: Two new methods of the fourth order for the simultaneous determination of multiple zeros of a polynomial are proposed. The presented methods are based on the fixed point relation of Laguerre's type and realized in ordinary complex arithmetic as well as circular complex interval arithmetic. The derived iterative formulas are suitable for the construction of modified methods with improved convergence rate with negligible additional operations. Very fast convergence of the considered methods is illustrated by two numerical examples. Keywords: Polynomial multiple zeros, simultaneous methods, inclusion of zeros, convergence. 1. INTRODUCTION The problem of solving nonlinear equations and systems of equations is one of the most important problems in the theory and practice, not only of applied mathematics including the theory of optimizations but also in many branches of engineering sciences, physics, computer science, astronomy, finance, and so on. As noted in [5], [6], [8], [9], [15], [16], [18], [23], Laguerre's method belongs to the most powerful methods for solving polynomial equations. The convergence characteristics of this method were extensively investigated in literature; references mentioned above are devoted to this subject. This method possesses local cubic convergence to a simple zero and excellent behavior in the case of polynomials with real zeros only. Two modifications of Laguerre's method, which enable simultaneous determination of all simple zeros of a polynomial and possess the convergence rate at least four, were proposed in [9]. In addition, the implementation on parallel computers and the comparison of these .