When Newton meets Descartes: A Simple and Fast Algorithm to Isolate the Real Roots of a Polynomial

TL;DR

The paper presents DSC2, a novel algorithm combining Descartes' Rule of Signs and Newton iteration to efficiently isolate real roots of polynomials with improved complexity and quadratic convergence.

Contribution

It introduces DSC2, a simple, fast, and easy-to-implement algorithm that improves the complexity bounds for real root isolation by integrating Newton steps with sign variation analysis.

Findings

Achieves quadratic convergence in most iterations.

Recursion tree size is almost optimal, O(nlog(n tau)).

Requires ilde{O}(n^3 tau) bit operations, matching the best known bounds.

Abstract

We introduce a new algorithm denoted DSC2 to isolate the real roots of a univariate square-free polynomial f with integer coefficients. The algorithm iteratively subdivides an initial interval which is known to contain all real roots of f. The main novelty of our approach is that we combine Descartes' Rule of Signs and Newton iteration. More precisely, instead of using a fixed subdivision strategy such as bisection in each iteration, a Newton step based on the number of sign variations for an actual interval is considered, and, only if the Newton step fails, we fall back to bisection. Following this approach, our analysis shows that, for most iterations, we can achieve quadratic convergence towards the real roots. In terms of complexity, our method induces a recursion tree of almost optimal size O(nlog(n tau)), where n denotes the degree of the polynomial and tau the bitsize of its coefficients. The latter bound constitutes an improvement by a factor of tau upon all existing subdivision methods for the task of isolating the real roots. In addition, we provide a bit complexity analysis showing that DSC2 needs only \tilde{O}(n^3tau) bit operations to isolate all real roots of f. This matches the best bound known for this fundamental problem. However, in comparison to the much more involved algorithms by Pan and Sch\"onhage (for the task of isolating all complex roots) which achieve the same bit complexity, DSC2 focuses on real root isolation, is very easy to access and easy to implement.