On the speed of convergence of Newton's method for complex polynomials

TL;DR

This paper presents an improved analysis of Newton's method for complex polynomials, providing explicit starting points and showing faster convergence with high probability for randomly distributed roots.

Contribution

It introduces a universal set of starting points for Newton's method that guarantees rapid convergence for all roots of degree d polynomials, improving previous worst-case bounds.

Findings

Number of iterations is O(d^2 log^4 d + d log|log ε|) with high probability

Explicit set of starting points of size ~3.33d log^2 d for all degree d polynomials

Improves upon previous worst-case iteration bounds for Newton's method

Abstract

We investigate Newton's method for complex polynomials of arbitrary degree $d$, normalized so that all their roots are in the unit disk. For each degree $d$, we give an explicit set $\mathcal{S}_d$ of $3.33d\log^2 d(1 + o(1))$ points with the following universal property: for every normalized polynomial of degree $d$ there are $d$ starting points in $\mathcal{S}_d$ whose Newton iterations find all the roots with a low number of iterations: if the roots are uniformly and independently distributed, we show that with probability at least $1-2/d$ the number of iterations for these $d$ starting points to reach all roots with precision $\varepsilon$ is $O(d^2\log^4 d + d\log|\log \varepsilon|)$. This is an improvement of an earlier result in \cite{Schleicher}, where the number of iterations is shown to be $O(d^4\log^2 d + d^3\log^2d|\log \varepsilon|)$ in the worst case (allowing multiple roots) and $O(d^3\log^2 d(\log d + \log \delta) + d\log|\log \varepsilon|)$ for well-separated (so-called $\delta$-separated) roots. Our result is almost optimal for this kind of starting points in the sense that the number of iterations can never be smaller than $O(d^2)$ for fixed $\varepsilon$.