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

The paper establishes global convergence guarantees for Backtracking New Q-Newton's method (BNQN) when applied to find roots of meromorphic functions in one complex variable, without requiring random parameters or genericity. It identifies an exceptional set \mathcal{E} of starting points that is locally contained in a countable union of real analytic curves and proves that BNQN trajectories converge to roots or infinity depending on the starting region; near critical points that are not roots, BNQN exhibits a structured d-fold saddle dynamics with 2d invariant curves. The work also provides a comprehensive local analysis of BNQN dynamics, derives approximate update formulas, and links the BNQN behavior to Newton's flow, Poincaré-Bendixson theory, and Voronoi diagrams, offering both rigorous results and heuristic explanations for observed experimental phenomena. These results deepen understanding of BNQN’s global behavior and offer practical insights for root-finding in complex meromorphic settings with minimal probabilistic assumptions.

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 $°(P)>°(Q)$, 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 Poincaré-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.

Figures (12)

• Figure 1: Basins of attraction for finding the 7 roots inside the domain $|z|<10$ of $f(z)=e^{2iz}-1$, an entire function with no critical points. Top picture: Newton's method, bottom picture: Random Relaxed Newton's methods. Points of the same colour belong to basin of attraction of the same root. Points in black do not belong to the basin of attraction of any of the concerned 7 roots. Note that the 2 pictures look very similar.
• Figure 2: Basins of attraction for finding the 7 roots inside the domain $|z|<10$ of $f(z)=e^{2iz}-1$, an entire function with no critical points. Top picture: Voronoi's diagram of the 7 roots, bottom picture: BNQN. Points of the same colour belong to basin of attraction of the same root. Points in black do not belong to the basin of attraction of any of the concerned 7 roots. Here $\theta =1$ in Algorithm \ref{['table:alg0']}. Note that the 2 pictures look very similar.
• Figure 3: An illustration of Theorem \ref{['Theorem2']}: The orbits of BNQN near a critical point of order $d=3$. Here the local dynamics is that of a 3-fold saddle point.
• Figure 4: The stable and unstable manifolds.
• Figure 5: The orbits of the linearization $\Phi _1$ near the critical point for $d=3$.

Theorems & Definitions (26)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Theorem 2.1: A convergence result for root finding
• Theorem 2.2: Invariant under appropriate linear changes of coordinates
• Theorem 2.3: Stable/Unstable manifolds for real analytic maps
• proof : Sketch of proof