New Combinations of Polynomial Root-Finding Iterations

Some near-optimal polynomial root-finders of 2024-25, based on subdivision iterations, approximate all complex roots of a polynomial or all roots in a fixed Region of Interest in the complex plane. The iterations can be applied to a black box polynomial, represented by an oracle (black box subroutine) for its evaluation rather than in monomial basis - by coefficients. We propose further empirical acceleration, for which we combine these iterations with Ehrlich's (aka Aberth's), Newton's, or Schroeder's. Our combinations of Ehrlich/Newton/Schroeder's and subdivision iterations can be applied to a black box polynomial and promises to support empirical acceleration versus each approach standing alone. A by-product of our study is a natural extension of the Gauss-Lucas theorem, of independent interest.