Newton's method in practice II: The iterated refinement Newton method and near-optimal complexity for finding all roots of some polynomials of very large degrees

TL;DR

This paper demonstrates a practical Newton's method-based algorithm capable of finding all roots of extremely high-degree polynomials (over one billion) efficiently, with observed complexity close to optimal, on standard desktop computers.

Contribution

It introduces a practical implementation of Newton's method for very large polynomials, achieving near-optimal complexity in root-finding for degrees exceeding one billion.

Findings

Successfully computed roots of polynomials with degree >10^9

Observed complexity between O(d ln d) and O(d ln^3 d)

Performed all computations on standard desktop computers

Abstract

We present a practical implementation based on Newton's method to find all roots of several families of complex polynomials of degrees exceeding one billion ($10^9$) so that the observed complexity to find all roots is between $O(d\ln d)$ and $O(d\ln^3 d)$ (measuring complexity in terms of number of Newton iterations or computing time). All computations were performed successfully on standard desktop computers built between 2007 and 2012.