用于求解单复变量亚纯函数根的回溯新Q-牛顿法：全局收敛性与局部稳定/不稳定曲线
Backtracking New Q-Newton's method for finding roots of meromorphic functions in 1 complex variable: Global convergence, and local stable/unstable curves
AI总结：
本文深入研究回溯新Q-牛顿法求解单复变量亚纯函数根的性质，证明了例外集的存在性及相关收敛结论，解释了该方法与牛顿法、庞加莱-本迪克松定理的动力学相似性，无需随机参数或通有函数假设。
摘要：
本文深入研究了由第三作者近期提出的回溯新Q-牛顿法（Backtracking New Q-Newton's method，简称BNQN）在求解亚纯函数根时的性质。若$f=P/Q$，其中$P$和$Q$是单复变量$z$的多项式，且$°(P)>°(Q)$，我们证明存在一个例外集$\text{mathcal{E}}\text{subset}\text{mathbf{C}}$，该集合包含于$\text{mathbf{R}}^2=\text{mathbf{C}}$中实解析曲线的可数并集内，使得以下命题A和B成立。其中，$\textbraceleft z_n\textbraceright$是由BNQN以非$f$极点的初始点$z_0$构造的序列。A) 若$z_0\text{in}\text{mathbf{C}}\text{backslash}\text{mathcal{E}}$，则$\textbraceleft z_n\textbraceright$收敛到$f$的一个根。B) 若$z_0\text{in} \text{mathcal{E}}$，则$\textbraceleft z_n\textbraceright$收敛到$f$的一个临界点而非根。实验似乎表明，通常情况下，即使$f$是多项式，集合$\text{mathcal{E}}$也不包含于实解析曲线的有限并集中。我们还提供了与$\text{mathcal{E}}$局部是否包含于有限条实解析曲线相关的进一步结果。类似结论对一般亚纯函数也成立。此外，与已有工作不同，本文不要求BNQN的参数是随机的，也不要求亚纯函数$f$是通有（generic）的。基于这些理论结果，我们从严格和启发式两个层面解释了作者此前工作中通过BNQN实验观察到的现象。特别是，BNQN（一种迭代方法）的动力学特性似乎与牛顿法（一种连续方法）以及复平面上可微实动力系统的经典庞加莱-本迪克松（Poincaré-Bendixon）定理存在惊人的相似性。考虑到离散版本的牛顿法（如松弛牛顿法）并不具备这种特性，这一发现更具研究价值。
英文摘要：
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.
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