论文标题
除了对根隔离算法的最坏情况分析之外
Beyond Worst-Case Analysis for Root Isolation Algorithms
论文作者
论文摘要
隔离单变量多项式的真正根源是符号计算中的一个基本问题，可以说是计算数学中最重要的问题之一。这个问题有悠久的历史，以许多巧妙的算法装饰，并提供了积极的研究领域。但是，根发现算法的最坏情况分析与它们的实际性能无关。我们为具有整数系数的多项式开发了平滑的分析框架，以弥合复杂性估计和实际性能之间的差距。 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 $\widetilde{\mathcal{O}}_B(d^2 + d τ)$, where $d$ is the degree of the polynomial and $τ$ the bitsize of the coefficients.
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 $\widetilde{\mathcal{O}}_B(d^2 + d τ)$, where $d$ is the degree of the polynomial and $τ$ the bitsize of the coefficients.
