The Root and Bell’s disk iteration methods are of the same error propagation characteristics in the simultaneous determination of the zeros of a polynomial, Part I: Correction methods

In this paper we consider the error propagation of the Root and Bell’s disk iteration methods enhanced by incorporating a correction term and a choice of a disk inversion formula in the methods, for the simultaneous computation of the zeros of a polynomial. The asymptotic error propagation is proved to be the same in both methods. This result is important considering the fact that these methods are in popular usage in the simultaneous computation of the zeros of a polynomial. The proof of the results herein follows the ideas of [M.S. Petkovic, C. Carstensen, Some improved inclusion methods for polynomial roots with Weierstrass corrections, Comput. Math. Appl. 25 (3) (1993) 59–67]. When the refinement process of correction is efficient, it is this mode of correction we have desired to propose.