We analyze different measures for the backward error of a set of numerical approximations for the roots of a polynomial. We focus mainly on the element-wise mixed backward error introduced by Mastronardi and Van Dooren, and the tropical backward error introduced by Tisseur and Van Barel. We show that these measures are equivalent under suitable assumptions. We also show relations between these measures and the classical element-wise and norm-wise backward error measures.

中文翻译：

---

多项式根的后向误差测度

我们分析多项式根的一组数值近似值的向后误差的不同度量。我们主要关注Mastronardi和Van Dooren引入的元素方式混合后向误差，以及Tisseur和Van Barel引入的热带后向误差。我们证明在适当的假设下这些措施是等效的。我们还显示了这些度量与经典的按元素和按规范的后向误差度量之间的关系。