Subdivision and Schröder's Polynomial Root-Finders Combined

Our root-finder at ACM-SIAM SODA 2024 accelerates classical subdivision iterations for approximation of all zeros of a polynomial as well as all its zeros lying in a fixed Region of Interest in the complex plane. The root-finder enhances the power of subdivision classical iterations and promises to compete for user's choice. At SODA 2024 we proved that that root-finder is near-optimal in some detail, except for its stage of near-maximal compression of a complex disc without losing its roots. Here we complete the proof and also propose to simplify that stage empirically by means of incorporation of Schröder's iterations.