# New Combinations of Polynomial Root-Finding Iterations

## TL;DR

This paper introduces new combinations of polynomial root-finding algorithms, enhancing empirical performance for complex roots, especially in black box scenarios, and extends the Gauss-Lucas theorem.

## Contribution

It proposes novel combinations of subdivision iterations with Ehrlich's and modified Newton's methods, improving empirical root-finding efficiency.

## Key findings

- Combined methods compete with MPSolve in accuracy.
- Applicable to black box polynomials and fixed regions.
- Extended Gauss-Lucas theorem offers theoretical insights.

## Abstract

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.