When Newton meets Descartes: a simple and fast algorithm to isolate the real roots of a polynomial

• Authors: Michael Sagraloff
• Venue: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
• Year: 2012

We introduce a novel algorithm denoted NewDsc 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 and performs exact (rational) operations on the coefficients of f in each step. For the subdivision strategy, 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, quadratic convergence towards the real roots is achieved in most iterations. 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. We further provide a detailed complexity analysis which shows that NewDsc needs only Õ(n3τ) bit operations to isolate all real roots of f. In comparison to existing asymptotically fast numerical algorithms (e.g. the algorithms by V. Pan and A. Schönhage), NewDsc is much easier to access and, due to its similarities to the classical Descartes method, it seems to be well suited for an efficient implementation.