Beyond Worst-Case Analysis for Root Isolation Algorithms

Abstract: Isolating the real roots of univariate polynomials is a fundamental problem in symbolic computation and it is arguably one of the most important problems in computational mathematics. The problem has a long history decorated with numerous ingenious algorithms and furnishes an active area of research. However, the worst-case analysis of root-finding algorithms does not correlate with their practical performance. We develop a smoothed analysis framework for polynomials with integer coefficients to bridge the gap between the complexity estimates and the practical performance. In this setting, we derive that the expected bit complexity of DESCARTES solver to isolate the real roots of a polynomial, with coefficients uniformly distributed, is $\widetilde{\mathcal{O}}_B(d^2 + d \tau)$, where $d$ is the degree of the polynomial and $\tau$ the bitsize of the coefficients.