Root-finding with eigren-solving

We survey and extend the recent progress in polynomial rootfinding via eigen-solving for highly structured generalized companion matrices. We cover the selection of eigen-solvers and matrices and show the benefits of exploiting matrix structure. No good estimates for the rate of global convergence of the eigen-solvers are known, but according to ample empirical evidence it is sufficient to use a constant number of iteration steps per eigenvalue. If so, the resulting root-finders are optimal up to a constant factor because they use linear arithmetic time per step and perform with a constant (double) precision. Some by-products of our study are of independent interest. The algorithms can be extended to solving secular equations.