Root Radii and Subdivision for Polynomial Root-Finding.

The recent subdivision algorithms for univariate polynomial Complex Root Clustering (CRC) and Real Root Isolation (RRI) approximate all roots in a fixed Region of Interest (RoI) and, like the algorithm of Pan (1995, 2002), achieve near optimal bit complexity for the so called benchmark problem. On the other hand, user's current choice for the approximation of all complex roots of a polynomial is the package MPSolve, implementing Ehrlich -- Aberth's iterations. Their very fast empirical global convergence (right from the start) is well-known although has no formal support. Subdivision iterations allow robust implementations, one of which is currently the user's choice for solving the RRI problem, including the task of the approximation of all real roots. Another implementation is slower than MPSolve (by several orders of magnitude) for finding all roots but outperforms MPSolve for solving the CRC problem where the RoI contains only a small number of roots. We analyze and extend a 2000 variant of Sch\"onhage's algorithm of (1982), which efficiently computes narrow real intervals that contain the moduli of all roots, thus defining a set of annuli covering all the complex roots. We present an implementable version of this algorithm and by using the computed sets of annuli improve in practice subdivision algorithms for CRC and RRI while supporting near optimal bit complexity.