Old and New Nearly Optimal Polynomial Root-Finders

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 and 2016, based on recursive factorization of a polynomial and subdivision iterations, respectively, but both of them are superseded in practice by Ehrlich's functional iterations.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. They match or supersede the known algorithms in terms of their estimated complexity for root-finding on the complex plane, in a disc, and in a line segment and promise to be practically competitive.