Real and complex polynomial root-finding with eigen-solving and preprocessing

Abstract

Recent progress on root-finding for polynomial and secular equations largely relied on eigen-solving for the associated companion and diagonal plus rank-one generalized companion matrices. By applying to them Rayleigh quotient iteration, we could have already competed with the current best polynomial root-finders, but we achieve further speedup by applying additive preprocessing. Moreover our novel rational maps of the input matrix enables us to direct the iteration to approximating only real roots, so that we dramatically accelerate their numerical computation in the important case where they are much less numerous than all complex roots.