SqFreeEVAL: An (almost) optimal real-root isolation algorithm

TL;DR

This paper introduces a simple, evaluation-based subdivision algorithm called SqFreeEVAL for isolating real roots of polynomials, providing an optimal complexity bound using novel continuous amortization analysis.

Contribution

It presents a new, simplified complexity analysis of the SqFreeEVAL algorithm using continuous amortization, achieving optimal bounds for root isolation.

Findings

Achieves an O(d(L+ln d)) bound on subdivision tree size.

First application of continuous amortization for this analysis.

Provides a simpler, optimal complexity bound for evaluation-based root isolation.

Abstract

Let f be a univariate polynomial with real coefficients, f in R[X]. Subdivision algorithms based on algebraic techniques (e.g., Sturm or Descartes methods) are widely used for isolating the real roots of f in a given interval. In this paper, we consider a simple subdivision algorithm whose primitives are purely numerical (e.g., function evaluation). The complexity of this algorithm is adaptive because the algorithm makes decisions based on local data. The complexity analysis of adaptive algorithms (and this algorithm in particular) is a new challenge for computer science. In this paper, we compute the size of the subdivision tree for the SqFreeEVAL algorithm. The SqFreeEVAL algorithm is an evaluation-based numerical algorithm which is well-known in several communities. The algorithm itself is simple, but prior attempts to compute its complexity have proven to be quite technical and have yielded sub-optimal results. Our main result is a simple O(d(L+ln d)) bound on the size of the subdivision tree for the SqFreeEVAL algorithm on the benchmark problem of isolating all real roots of an integer polynomial f of degree d and whose coefficients can be written with at most L bits. Our proof uses two amortization-based techniques: First, we use the algebraic amortization technique of the standard Mahler-Davenport root bounds to interpret the integral in terms of d and L. Second, we use a continuous amortization technique based on an integral to bound the size of the subdivision tree. This paper is the first to use the novel analysis technique of continuous amortization to derive state of the art complexity bounds.