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：一种名为回溯新Q牛顿法（BNQN）的牛顿法变体——具有强理论保证、易于实现且实验性能优异——近年来由第三作者提出。先前进行的实验表明，BNQN在求解多项式与亚纯函数根时吸引域展现出显著特性。总体而言，其吸引域相较于牛顿法更为光滑。本文继续深入实验探究这一显著现象，并将BNQN与牛顿流及沃罗诺伊图建立联系。这一关联引出了若干亟待解释的挑战性难题。实验还表明，相较于牛顿法与随机松弛牛顿法，BNQN对随机扰动具有更强的鲁棒性。