Beyond Worst-Case Analysis for Root Isolation Algorithms

TL;DR

This paper introduces a smoothed analysis framework for root isolation algorithms, showing that the expected complexity of the DESCARTES solver is much better aligned with practical performance than worst-case estimates.

Contribution

It develops a smoothed analysis approach for polynomial root-finding, providing realistic complexity estimates that bridge the gap between theory and practice.

Findings

Expected bit complexity of DESCARTES solver is O_B(d^2 + d au) for polynomials with uniformly distributed coefficients.

Smoothed analysis offers a more accurate complexity measure than worst-case analysis.

The framework explains the practical efficiency of 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.