We introduce a new algorithm denoted DSC 2 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(n · log(nτ)), where n denotes the degree of the polynomial and τ the bitsize of its coefficients. The latter bound constitutes an improvement by a factor of τ upon all existing subdivision methods for the task of isolating the real roots. In addition, we provide a bit complexity analysis showing that DSC 2 needs onlyÕ(n 3 τ) bit operations 1 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 Schönhage (for the task of isolating all complex roots) which achieve the same bit complexity, DSC 2 focuses on real root isolation, is very easy to access and easy to implement.
Sagraloff REF showed how to traverse such chains more efficiently by combining Descartes' Rule of Signs, bisection, and Newton iteration.
When Newton meets Descartes: A Simple and Fast Algorithm to Isolate the Real Roots of a Polynomial
ArXiv
ArXiv
Mathematics Computer Science