A geometric approach to approximating the limit set of eigenvalues for banded Toeplitz matrices
Teodor Bucht, Jacob S. Christiansen
TL;DR
The paper introduces a geometric framework to approximate the limit set $\Lambda(b)$ for banded Toeplitz matrices by transforming the infinite-intersection characterization $\Lambda(b)=\bigcap_{\rho>0} \mathrm{sp}\,T(b_{\rho})$ into a computable sequence of polygon intersections over a compact $[\rho_l,\rho_h]$ and polygonal spectral models $\mathrm{sp}\,T(b_{\rho}^D)$. It proves convergence in the Hausdorff metric, provides a method to obtain an approximating superset with guaranteed containment, and combines these with an algebraic subset to yield explicit Hausdorff-error bounds. The implementation relies on efficient polygon intersection/offsetting (via PyClipper/Clipper2) and introduces area-sweep improvements to select informative $\rho$-samples, achieving a practical $O(n^2 + mn\log m)$ time complexity on average. Empirical tests show the geometric approach performs comparably or better than existing algebraic methods, particularly for high-degree symbols, and quantifies convergence rates as $O(1/\sqrt{k})$ with respect to the number of basic operations. Overall, the work provides explicit error control and scalable computation for $\Lambda(b)$, with clear paths for optimization and extension to pseudospectral considerations.
Abstract
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.