Near optimal subdivision algorithms for real root isolation

Abstract

Isolating real roots of a square-free polynomial in a given interval is a fundamental problem in computational algebra. Subdivision based algorithms are a standard approach to solve this problem. For instance, Sturm's method, or various algorithms based on the Descartes's rule of signs. For the benchmark problem of isolating all the real roots of a polynomial of degree n and root separation σ, the size of the subdivision tree of most of these algorithms is bounded by O(log⁡1/σ) (assume σ<1). Moreover, it is known that this is optimal for subdivision algorithms that perform uniform subdivision, i.e., the width of the interval decreases by some constant. Recently Sagraloff (2012) and Sagraloff–Mehlhorn (2016) have developed algorithms for real root isolation that combine subdivision with Newton iteration to reduce the size of the subdivision tree to O(n(log⁡(nlog⁡1/σ))).We describe a subroutine that reduces the size of the subdivision tree of any subdivision algorithm for real root isolation. The subdivision tree size of our algorithm using predicates based on either the Descartes's rule of signs or Sturm sequences is bounded by O(nlog⁡n), which is close to the optimal value of O(n). The corresponding bound for the algorithm EVAL, which uses certain interval-arithmetic based predicates, is O(n2log⁡n). Our analysis differs in two key aspects from earlier approaches. First, we use the general technique of continuous amortization from Burr–Krahmer–Yap (2009), and extend it to handle non-uniform subdivisions; second, we use the geometry of clusters of roots instead of root bounds. The latter aspect enables us to derive a bound on the subdivision tree that is independent of the root separation σ. The number of Newton iterations is bounded by O(nlog⁡log⁡(1/σ)).