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 $\Lambda(b) = \cap_{\rho \in (0, \infty)} \text{sp } T(b_\rho)$. 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 $\Lambda(b)$. 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矩阵极限集的求解问题。主要结果是一种逼近极限集$\Lambda(b)$的新方法，其中$b$为带状Toeplitz矩阵的符号函数。该方法基于几何框架与公式$\Lambda(b) = \cap_{\rho \in (0, \infty)} \text{sp } T(b_\rho)$。我们证明了全交集可通过有限个$\rho$值的交集逼近，且通过$\text{sp } T(b_\rho)$的多边形近似解的交集，可获得在Hausdorff度量下收敛于$\Lambda(b)$的逼近多边形。进一步研究表明，可适当扩展$\text{sp } T(b_\rho)$的多边形近似解以确保其包含$\text{sp } T(b_\rho)$。由此得到的交集形成$\Lambda(b)$的逼近超集，该超集在Hausdorff度量下收敛于$\Lambda(b)$且严格包含$\Lambda(b)$。我们采用Python实现该算法并完成测试，其性能与现有算法相当，部分情形更优。本文论证（未经严格证明）算法平均时间复杂度为$O(n^2 + mn\log m)$，其中$n$为$\rho$值数量，$m$为逼近$\text{sp } T(b_\rho)\)的多边形顶点数。此外，我们论证了$\Lambda(b)$与逼近多边形及逼近超集的距离，对于$\Lambda(b)$的大多数区域以$O(1/\sqrt{k})$量级递减，其中$k$为算法所需的初等运算次数。