逼近带状 Toeplitz 矩阵特征值极限集的一种几何方法
A geometric approach to approximating the limit set of eigenvalues for banded Toeplitz matrices
AI总结：
本文提出基于有限缩放因子与谱多边形交集的几何方法，构造收敛且保证包含带状 Toeplitz 矩阵特征值极限集的逼近超集，并给出误差界与 Python 验证。
摘要：
本文研究如何求带状 Toeplitz 矩阵的极限集。我们的主要结果是提出一种逼近极限集 $Λ(b)$ 的新方法，其中 $b$ 是带状 Toeplitz 矩阵的符号。该新方法是几何性的，并基于公式 $Λ(b) = \cap_{ρ\in (0,\infty)} \text{sp } T(b_ρ)$，其中 $ρ$ 是缩放因子，即 $b_ρ(t):=b(ρt)$，而 $\text{sp}(\cdot)$ 表示谱。我们证明了整个交集可以用有限多个 $ρ$ 对应的交集来逼近，并且对 $\text{sp } T(b_ρ)$ 的多边形逼近取交集，可得到 $Λ(b)$ 的一个逼近多边形，该多边形在 Hausdorff 度量下收敛于 $Λ(b)$。此外，我们证明可以略微放大 $\text{sp } T(b_ρ)$ 的多边形逼近，以确保它们包含 $\text{sp } T(b_ρ)$。随后取交集即可得到 $Λ(b)$ 的一个逼近超集，它在 Hausdorff 度量下收敛于 $Λ(b)$，并保证包含 $Λ(b)$。将已有的代数（求根）方法与我们的逼近超集相结合，我们能够给出到真实极限集的 Hausdorff 距离的显式界。我们用 Python 实现并测试了该算法。其性能与现有算法相当，在某些情况下更优。我们论证但未证明，该算法的平均时间复杂度为 $O(n^2+mn\log m)$，其中 $n$ 是 $ρ$ 的数量，$m$ 是逼近 $\text{sp } T(b_ρ)$ 的多边形顶点数。此外，我们论证，对于 $Λ(b)$ 的大部分区域，$Λ(b)$ 到逼近多边形和逼近超集的距离以 $O(1/\sqrt{k})$ 减小，其中 $k$ 是算法所需的基本运算次数。
英文摘要：
This article is about finding the limit set for banded Toeplitz matrices. Our main result is a new approach to approximate the limit set $Λ(b)$ where $b$ is the symbol of the banded Toeplitz matrix. The new approach is geometrical and based on the formula $Λ(b) = \cap_{ρ\in (0, \infty)} \text{sp } T(b_ρ)$, where $ρ$ is a scaling factor, i.e. $b_ρ(t) := b(ρt)$, and $\text{sp }(\cdot)$ denotes the spectrum. We show that the full intersection can be approximated by the intersection for a finite number of $ρ$'s, and that the intersection of polygon approximations for $\text{sp } T(b_ρ)$ yields an approximating polygon for $Λ(b)$ that converges to $Λ(b)$ in the Hausdorff metric. Further, we show that one can slightly expand the polygon approximations for $\text{sp } T(b_ρ)$ to ensure that they contain $\text{sp } T(b_ρ)$. Then, taking the intersection yields an approximating superset of $Λ(b)$ which converges to $Λ(b)$ in the Hausdorff metric, and is guaranteed to contain $Λ(b)$. 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 $ρ$'s and $m$ is the number of vertices for the polygons approximating $\text{sp } T(b_ρ)$. Further, we argue that the distance from $Λ(b)$ to both the approximating polygon and the approximating superset decreases as $O(1/\sqrt{k})$ for most of $Λ(b)$, where $k$ is the number of elementary operations required by the algorithm.
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