发表机构
University of Missouri（密苏里大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种基于围道积分表示的蒙特卡洛估计器，用于无偏估计孤立多项式零点，并给出方差与概率误差界，数值实验验证其有效性。
AI 中文摘要
我们提出了一种蒙特卡洛方法，通过围道积分表示来估计孤立多项式零点。对于被隔离围道包围的简单零点，我们证明该零点可以精确表示为通过对围道均匀采样获得的复值随机变量的期望。这产生了一个无偏估计器，且无需扰动多项式系数。我们建立了其方差，并基于浓度不等式推导了有限样本概率误差界，这些界明确反映了围道的几何形状及其与零点的分离程度。该框架还提供了基于围道的根计数和根隔离的随机公式。数值结果展示了估计器在不同多项式次数下的行为，并且我们进一步提供了与现有方法的比较。
英文摘要
We introduce a Monte Carlo approach for estimating isolated polynomial zeros through their contour integral representations. For a simple zero enclosed by an isolating contour, we show that the zero can be expressed exactly as the expectation of a complex-valued random variable obtained by uniformly sampling the contour. This yields an unbiased estimator without perturbing the polynomial coefficients. We establish its variance and derive finite sample probabilistic error bounds based on concentration inequalities, with the bounds explicitly reflecting the geometry of the contour and its separation from the zeros. The framework also provides a stochastic formulation of contour based root counting and root isolation. Numerical results illustrate the estimator's behavior under varying polynomial degrees, and we further provide comparisons to existing methods.
Comments21 pages, 2 plots, 1 table, Keywords: Argument Principle, Cauchy Integral, Contour Integration, Hoeffding Error Bounds, Monte Carlo Integration, Monte Carlo Simulation, Polynomial Roots
