New Combinations of Polynomial Root-Finding Iterations

Abstract

Some near-optimal polynomial root-finders of 2024-25, based on subdivision iterations, approximate all complex roots of a polynomial or all roots lying in a fixed Region of Interest in the complex plane. We combine these iterations with Newton's and/or Schroeder's to yield significant empirical acceleration versus each approach standing alone. Like the cited recent algorithms, our root-finders can be applied not only to a polynomial represented in monomial basis by its coefficients but also to a black box polynomial represented by an oracle (black box subroutine) for its evaluation. Some by-products of our study such as an extension of the Gauss-Lucas theorem and a fast black box estimator for root radius can be of independent interest.