Backtracking New Q-Newton's method, Newton's flow, Voronoi's diagram and Stochastic root finding
This work addresses the problem of improving root-finding algorithms in numerical analysis, but it is incremental as it builds on a recently introduced method with experimental exploration.
The paper explores the Backtracking New Q-Newton's method (BNQN), a variant of Newton's method, by experimentally investigating its smooth basins of attraction for finding roots of polynomials and meromorphic functions, and connects it to Newton's flow and Voronoi's diagrams, while noting its robustness against random perturbations compared to other methods.
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.