Given a black box (oracle) for the evaluation of a univariate polynomial p(x) of a degree d, we seek its zeros, that is, the roots of the equation p(x)=0. At FOCS 2016 Louis and Vempala approximated a largest zero of such a real-rooted polynomial within $1/2^b$, by performing at NR cost of the evaluation of Newton's ratio p(x)/p'(x) at O(b\log(d)) points x. They readily extended this root-finder to record fast approximation of a largest eigenvalue of a symmetric matrix under the Boolean complexity model. We apply distinct approach and techniques to obtain more general results at the same computational cost.

翻译：给定一个用于评估d次单变量多项式p(x)的黑箱（预言机），我们寻求其零点，即方程p(x)=0的根。在FOCS 2016会议上，Louis和Vempala通过在O(b\log(d))个点x上以牛顿比值p(x)/p'(x)的NR代价进行评估，将此类实根多项式的最大零点近似到$1/2^b$以内。他们将该求根器直接扩展，在布尔复杂度模型下记录对称矩阵最大特征值的快速近似。我们采用不同的途径和技术，以相同的计算代价获得更一般的结果。