Root Refinement for Real Polynomials

TL;DR

This paper introduces a certified, adaptive algorithm for refining real roots of square-free polynomials to high precision, combining quadratic interval refinement with robustness to coefficient inaccuracies, and provides near-optimal complexity bounds.

Contribution

It adapts quadratic interval refinement to handle coefficient inaccuracies, providing a simple, practical, and near-optimal root refinement algorithm with theoretical guarantees.

Findings

Algorithm refines roots to $2^{-L}$ precision efficiently.

Bit complexity bounds are near optimal and improve previous results.

Practical performance closely matches theoretical asymptotic bounds.

Abstract

We consider the problem of approximating all real roots of a square-free polynomial $f$. Given isolating intervals, our algorithm refines each of them to a width of $2^{-L}$ or less, that is, each of the roots is approximated to $L$ bits after the binary point. Our method provides a certified answer for arbitrary real polynomials, only considering finite approximations of the polynomial coefficients and choosing a suitable working precision adaptively. In this way, we get a correct algorithm that is simple to implement and practically efficient. Our algorithm uses the quadratic interval refinement method; we adapt that method to be able to cope with inaccuracies when evaluating $f$, without sacrificing its quadratic convergence behavior. We prove a bound on the bit complexity of our algorithm in terms of the degree of the polynomial, the size and the separation of the roots, that is, parameters exclusively related to the geometric location of the roots. Our bound is near optimal and significantly improves previous work on integer polynomials. Furthermore, it essentially matches the best known theoretical bounds on root approximation which are obtained by very sophisticated algorithms. We also investigate the practical behavior of the algorithm and demonstrate how closely the practical performance matches our asymptotic bounds.