A new variant of Newton's method - named Backtracking New Q-Newton's method (BNQN) - which has strong theoretical guarantee, is easy to implement, and has good experimental performance, was recently introduced by the third author. Experiments performed previously showed some remarkable properties of the basins of attractions for finding roots of polynomials and meromorphic functions, with BNQN. In general, they look more smooth than that of Newton's method. In this paper, we continue to experimentally explore in depth this remarkable phenomenon, and connect BNQN to Newton's flow and Voronoi's diagram. This link poses a couple of challenging puzzles to be explained. Experiments also indicate that BNQN is more robust against random perturbations than Newton's method and Random Relaxed Newton's method.

翻译：一种名为回溯式新牛顿法（BNQN）的牛顿法新变体——该变体具有强理论保证、易于实现且实验性能良好——近期由第三作者提出。先前实验表明，BNQN在多项式与亚纯函数求根过程中，其吸引盆展现出若干显著特性。总体而言，这些吸引盆比牛顿法对应的吸引盆更为平滑。本文通过深入实验研究这一显著现象，并将BNQN与牛顿流及沃罗诺伊图建立关联。这一关联提出了若干具有挑战性的待解谜题。实验同时表明，BNQN对随机扰动的鲁棒性优于牛顿法和随机松弛牛顿法。