We investigate two well known dynamical systems that are designed to find roots of univariate polynomials by iteration: the methods known by Newton and by Ehrlich–Aberth. Both are known to have found all roots of high degree polynomials with good complexity. Our goal is to determine in which cases which of the two algorithms is more efficient. We come to the conclusion that Newton is faster when the polynomials are given by recursion so they can be evaluated in logarithmic time with respect to the degree, or when all the roots are all near the boundary of their convex hull. Conversely, Ehrlich–Aberth has the advantage when no fast evaluation of the polynomials is available, and when roots are in the interior of the convex hull of other roots.

中文翻译：

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通过动力学系统找到多项式根–案例研究

我们研究了两个众所周知的动力学系统，这些系统旨在通过迭代查找单变量多项式的根：Newton和Ehrlich-Aberth所熟知的方法。众所周知，两者都发现了具有高复杂度的高次多项式的所有根。我们的目标是确定两种算法中哪种算法效率更高。我们得出的结论是，当多项式通过递归给出时，牛顿速度更快，因此可以相对于度数以对数形式评估它们，或者当所有根都位于其凸包的边界附近时。相反，当无法快速评估多项式时，并且当根在其他根的凸包内部时，Ehrlich-Aberth具有优势。