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We propose a new simple but nearly optimal algorithm for the approximation ofall sufficiently well isolated complex roots and root clusters of a univariatepolynomial. Quite typically the known root-finders at first compute some crudebut reasonably good approximations to well-conditioned roots (that is, thoseisolated from the other roots) and then refine the approximations very fast, byusing Boolean time which is nearly optimal, up to a polylogarithmic factor. Bycombining and extending some old root-finding techniques, the geometry of thecomplex plane, and randomized parametrization, we accelerate the initial stageof obtaining crude to all well-conditioned simple and multiple roots as well asisolated root clusters. Our algorithm performs this stage at a Boolean costdominated by the nearly optimal cost of subsequent refinement of theseapproximations, which we can perform concurrently, with minimum processorcommunication and synchronization. Our techniques are quite simple andelementary; their power and application range may increase in their combinationwith the known efficient root-finding methods.

Simple and Nearly Optimal Polynomial Root-finding by Means of Root Radii Approximation