Nearly Optimal Hybrid Polynomial Root-finders
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Univariate polynomial root-finding has been studied for four millennia and still remains the subject of intensive research. Hundreds if not thousands of efficient algorithms for this task have been proposed and analyzed. One of them, of 1982, by Schönhage, advancing the one of 1967 by Delves and Lyness, splits an input polynomial into two factors and then recursively splits the factors similarly until it approximates factorization of an input polynomial into the product of linear factors and recovers all roots from these factors. By further advancing this algorithm in 1995 we approximated all roots in record and nearly optimal Boolean time. Our construction is quite involved and is hard to implement, but the polynomial splitting sub-algorithm is simpler and practically promising. We describe and briefly analyze hybrid algorithms that combine it with either functional or subdivision iterations. Both of these iterations are most popular and nearly optimal polynomial root-finders, but the hybrid algorithms should further enhance their efficiency for the approximation of all roots of a polynomial on the complex plane as well as its roots in a disc and, with incorporation of some novelties, in a line interval.
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