Old and New Nearly Optimal Polynomial Root-finders

05/30/2018

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by   Victor Y. Pan, et al.

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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. Two nearly optimal solution algorithms have been devised in 1995, based on recursive factorization of a polynomial, and in 2016, based on subdivision iterations, but both of them are superseded in practice by Ehrlich's functional iterations. By combining factorization techniques with Ehrlich's and subdivision iterations we devise a variety of new root-finders that run at the same or lower Boolean cost and promise to compete with both classes of iterations for approximation of all complex roots of a polynomial and its roots in a disc on the complex plain. Further simple novelties lead us to a competitive root-finder in a line segment.