Reliable Computing 10: 437–467, 2004.
2004 Kluwer Academic Publishers. Printed in the Netherlands.
Ostrowski-like Method with Corrections
for the Inclusion of Polynomial Zeros
MIODRAG S. PETKOVI
SAN M. MILO
Faculty of Electronic Engineering, University of Ni
s, P.O. Box 73, 18 000 Ni
s, Serbia and
Montenegro, e-mail: firstname.lastname@example.org
(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.
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  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. , ).
First results on iterative interval methods for the simultaneous approximation of
polynomial zeros were established in , , . An extensive study and history
of interval methods for solving algebraic equations may be found in the books ,
, and . 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 .
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