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Finding polynomial roots by dynamical systems -- a case study

We investigate two well known dynamical systems that are designed to find roots of univariate polynomials by iteration: the methods known by Newton and by Ehrlich-Aberth. Both are known to have found all roots of high degree polynomials with good complexity. Our goal is to determine in which cases which of the two algorithms is more efficient. We come to the conclusion that Newton is faster when the polynomials are given by recursion so they can be evaluated in logarithmic time with respect to the degree, or when all the roots are all near the boundary of their convex hull. Conversely, Ehrlich-Aberth has the advantage when no fast evaluation of the polynomials is available, and when roots are in the interior of the convex hull of other roots.