No abstract

No abstract

No abstract

Abstract

The problem of isolating real roots of a square-free polynomial inside a given interval I0 is a fundamental problem. Subdivision based algorithms are a standard approach to solve this problem. Given an interval I, such algorithms rely on two predicates: an exclusion predicate, which if true means I has no roots, and an inclusion predicate, which if true, reports an isolated root in I. If neither predicate holds, then we subdivide the interval and proceed recursively, starting from I0. Example algorithms are Sturm's method (predicates based on Sturm sequences), the Descartes method (using Descartes's rule of signs), and Eval (using interval-arithmetic). For the canonical problem of isolating all real roots of a degree n polynomial with integer coefficients of bit-length L, the subdivision tree size of (almost all) these algorithms is bounded by O(n(L + log n)). This is known to be optimal for subdivision based algorithms.We describe a subroutine that improves the running time of any subdivision algorithm for real root isolation. The subroutine first detects clusters of roots using a result of Ostrowski, and then uses Newton iteration to converge to them. Near a cluster, we switch to subdivision, and proceed recursively. The subroutine has the advantage that it is independent of the predicates used to terminate the subdivision. This gives us an alternative and simpler approach to recent developments of Sagraloff (2012) and SagraloffMehlhorn (2013), assuming exact arithmetic.The subdivision tree size of our algorithm using predicates based on Descartes's rule of signs is bounded by O(n log n), which is better by O(n log L) compared to known results. Our analysis differs in two key aspects. First, we use the general technique of continuous amortization from Burr-Krahmer-Yap (2009), and second, we use the geometry of clusters of roots instead of the Davenport-Mahler bound. The analysis naturally extends to other predicates.