Reliable Computing 10: 437–467, 2004. 437 c  2004 Kluwer Academic Publishers. Printed in the Netherlands. Ostrowski-like Method with Corrections for the Inclusion of Polynomial Zeros MIODRAG S. PETKOVI ´ C and DU ˇ SAN M. MILO ˇ SEVI ´ C Faculty of Electronic Engineering, University of Niˇ s, P.O. Box 73, 18 000 Niˇ s, Serbia and Montenegro, e-mail: msp@junis.ni.ac.yu (Received: 26 July 2003; accepted: 20 January 2004) Abstract. In this paper we construct iterative methods of Ostrowski’s type for the simultaneous inclusion of all zeros of a polynomial. Using the concept of the R-order of convergence of mutually dependent sequences, we present the convergence analysis of the total-step and the single-step methods with Newton and Halley’s corrections. The case of multiple zeros is also considered. The suggested algorithms possess a great computational efficiency since the increase of the convergence rate is attained without additional calculations. Numerical examples and an analysis of computational efficiency are given. 1. Introduction Iterative methods for the simultaneous determination of polynomial zeros, realized in interval arithmetic, produce resulting real or complex intervals (disks or rectan- gles) containing the wanted zeros. In this manner an information about upper error bounds of approximations to the zeros are provided (see [16] for more details). Moreover, in some practical problems of applied and industrial mathematics, alge- braic polynomials with uncertain coefficients can appear. This kind of problems is effectively solved applying interval methods (c.f. [19], [20]). First results on iterative interval methods for the simultaneous approximation of polynomial zeros were established in [2], [6], [7]. An extensive study and history of interval methods for solving algebraic equations may be found in the books [1], [16], and [20]. The purpose of this paper is to present Ostrowski-like algorithms for the simultaneous inclusion of the zeros of a polynomial. These algorithms are realized in circular complex arithmetic and can be regarded as modifications of the fourth order method proposed by Gargantini [6]. This paper is organized as follows. The basic properties of circular complex arithmetic, necessary for the development and convergence analysis of the present- ed inclusion methods, are given in Section 2. In Section 3 we give the fixed-point relation of Ostrowski-type which makes the base for the construction of simulta- neous Ostrowski-like interval methods. The derivation of the basic fourth order method and the criterion for the choice of a proper square root of a disk are given in Section 3. The modified total-step method with the increased convergence speed is developed in Section 4 using Newton’s and Halley’s correction. The convergence