New Combinations of Polynomial Root-Finding Iterations
For researchers in numerical analysis, this work offers incremental improvements to polynomial root-finding by combining existing methods, but lacks empirical validation.
The paper combines near-optimal polynomial root-finders based on subdivision iterations with Ehrlich's and modified Newton's methods to accelerate root-finding for black box polynomials. The combinations promise empirical acceleration over each method alone, though no concrete numbers are provided.
The 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 and 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. Towards further empirical acceleration we combine them with two other popular root-finders, Ehrlich's (aka Aberth's) and modified Newton's. They have weaker formal support but compete empirically with user's current choice root-finder MPSolve for approximation of all complex roots under proper initialization that involve polynomial coefficients. Their combinations with subdivision iterations can be applied to a black box polynomial and to root-finding in a fixed Region and promise to support empirical acceleration versus each approach standing alone. For a by-product of our study, we naturally extend the Gauss-Lucas theorem; this can be of independent interest.