Near optimal tree size bounds on a simple real root isolation algorithm

The problem of isolating all real roots of a square-free integer polynomial f(X) inside any given interval I₀ is a fundamental problem. EVAL is a simple and practical exact numerical algorithm for this problem: it recursively bisects I0, and any sub-interval I ⊆ I0, until a certain numerical predicate C₀(I) ⊆ C₁(I) holds on each I. We prove that the size of the recursion tree is O(d(L + r + log d)) where f has degree d, its coefficients have absolute values < 2^L, and I0 contains r roots of f. In the range L ≥ d, our bound is the sharpest known, and provably optimal. Our results are closely paralleled by recent bounds on EVAL by Sagraloff-Yap (ISSAC 2011) and Burr-Krahmer (2012). In the range L ≤ d, our bound is incomparable with those of Sagraloff-Yap or Burr-Krahmer. Similar to the Burr-Krahmer proof, we exploit the technique of "continuous amortization" from Burr-Krahmer-Yap (2009), namely to bound the tree size by an integral I₀ G(x)dx over a suitable "charging function" G(x). We give an application of this feature to the problem of ray-shooting (i.e., finding smallest root in a given interval).