Backtracking New Q-Newton's method for finding roots of meromorphic functions in 1 complex variable: Global convergence, and local stable/unstable curves

TL;DR

This paper analyzes the properties of Backtracking New Q-Newton's method for finding roots of meromorphic functions, establishing convergence outside a complex exceptional set and exploring the structure of this set.

Contribution

The paper provides new theoretical insights into the global convergence and structure of the exceptional set for Backtracking New Q-Newton's method without requiring randomness or generic conditions.

Findings

Convergence to roots for initial points outside a countable union of real analytic curves.

The exceptional set is contained in a countable union of real analytic curves.

Similar dynamics to Newton's method and Poincaré-Bendixson theorem observed.

Abstract

In this paper, we research more in depth properties of Backtracking New Q-Newton's method (recently designed by the third author), when used to find roots of meromorphic functions. If $f=P/Q$, where $P$ and $Q$ are polynomials in 1 complex variable z with $\deg (P)>\deg (Q)$, we show the existence of an exceptional set $\mathcal{E}\subset\mathbf{C}$, which is contained in a countable union of real analytic curves in $\mathbf{R}^2=\mathbf{C}$, so that the following statements A and B hold. Here, $\{z_n\}$ is the sequence constructed by BNQN with an initial point $z_0$ which is not a pole of $f$. A) If $z_0\in\mathbf{C}\backslash\mathcal{E}$, then $\{z_n\}$ converges to a root of $f$. B) If $z_0\in \mathcal{E}$, then $\{z_n\}$ converges to a critical point - but not a root - of $f$. Experiments seem to indicate that in general, even when $f$ is a polynomial, the set $\mathcal{E}$ is not contained in a finite union of real analytic curves. We provide further results relevant to whether locally $\mathcal{E}$ is contained in a finite number of real analytic curves. A similar result holds for general meromorphic functions. Moreover, unlike previous work, here we do not require that the parameters of BNQN are random, or that the meromorphic function $f$ is generic. Based on the theoretical results, we explain (both rigorously and heuristically) of what observed in experiments with BNQN, in previous works by the authors. In particular, the dynamics of BNQN (an iterative method) seems to have some striking similarities to Newton's method (a continuous method) and the classical Poincar\'e-Bendixon theorem for differentiable real dynamical systems on the complex plane. This is the more interesting given that discrete versions of Newton's method (e.g. Relaxed Newton's method) does not behave this way.