This article is about finding the limit set for banded Toeplitz matrices. Our main result is a new approach to approximate the limit set $\Lambda(b)$ where $b$ is the symbol of the banded Toeplitz matrix. The new approach is geometrical and based on the formula $\Lambda(b) = \cap_{\rho \in (0, \infty)} \text{sp } T(b_\rho)$, where $\rho$ is a scaling factor, i.e. $b_\rho(t) := b(\rho t)$, and $\text{sp }(\cdot)$ denotes the spectrum. We show that the full intersection can be approximated by the intersection for a finite number of $\rho$'s, and that the intersection of polygon approximations for $\text{sp } T(b_\rho)$ yields an approximating polygon for $\Lambda(b)$ that converges to $\Lambda(b)$ in the Hausdorff metric. Further, we show that one can slightly expand the polygon approximations for $\text{sp } T(b_\rho)$ to ensure that they contain $\text{sp } T(b_\rho)$. Then, taking the intersection yields an approximating superset of $\Lambda(b)$ which converges to $\Lambda(b)$ in the Hausdorff metric, and is guaranteed to contain $\Lambda(b)$. Combining the established algebraic (root-finding) method with our approximating superset, we are able to give an explicit bound on the Hausdorff distance to the true limit set. We implement the algorithm in Python and test it. It performs on par to and better in some cases than existing algorithms. We argue, but do not prove, that the average time complexity of the algorithm is $O(n^2 + mn\log m)$, where $n$ is the number of $\rho$'s and $m$ is the number of vertices for the polygons approximating $\text{sp } T(b_\rho)$. Further, we argue that the distance from $\Lambda(b)$ to both the approximating polygon and the approximating superset decreases as $O(1/\sqrt{k})$ for most of $\Lambda(b)$, where $k$ is the number of elementary operations required by the algorithm.

翻译：本文研究带状Toeplitz矩阵的极限集求解问题。我们提出了一种逼近极限集$\Lambda(b)$的新方法，其中$b$为带状Toeplitz矩阵的符号函数。该方法基于几何原理，利用公式$\Lambda(b) = \cap_{\rho \in (0, \infty)} \text{sp } T(b_\rho)$，其中$\rho$为缩放因子（即$b_\rho(t) := b(\rho t)$），$\text{sp }(\cdot)$表示谱集。我们证明：完整交集可通过有限个$\rho$值的交集进行逼近；通过对$\text{sp } T(b_\rho)$的多边形逼近取交集，可获得$\Lambda(b)$的逼近多边形，该多边形在Hausdorff度量意义下收敛于$\Lambda(b)$。进一步研究表明，可对$\text{sp } T(b_\rho)$的多边形逼近进行适度扩展，确保其包含$\text{sp } T(b_\rho)$。此时取交集将得到$\Lambda(b)$的逼近超集，该超集在Hausdorff度量意义下收敛于$\Lambda(b)$，且严格包含$\Lambda(b)$。将已有的代数方法（求根法）与本文提出的逼近超集相结合，我们能够给出逼近结果与真实极限集之间Hausdorff距离的显式上界。我们在Python中实现了该算法并进行测试，其性能与现有算法相当，在某些情况下更优。我们通过论证（未严格证明）指出：算法的平均时间复杂度为$O(n^2 + mn\log m)$，其中$n$为$\rho$值的数量，$m$为逼近$\text{sp } T(b_\rho)$的多边形顶点数。此外，我们论证了对于$\Lambda(b)$的大部分区域，逼近多边形和逼近超集与$\Lambda(b)$的距离以$O(1/\sqrt{k})$的速率递减，其中$k$为算法所需的基本运算次数。