# A geometric approach to approximating the limit set of eigenvalues for banded Toeplitz matrices

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) = ∩_ρ∈ (0, ∞)sp T(b_ρ). 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 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 sp T(b_ρ) to ensure that they contain 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). 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 + mnlog m), where n is the number of ρ's and m is the number of vertices for the polygons approximating 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/√(k)) for most of Λ(b), where k is the number of elementary operations required by the algorithm.

• 1 publication
• 1 publication
research
07/16/2023

### Better Diameter Algorithms for Bounded VC-dimension Graphs and Geometric Intersection Graphs

We develop a framework for algorithms finding diameter in graphs of boun...
research
09/21/2018

### A numerical approach for the filtered generalized Cech complex

In this paper, we present an algorithm to compute the filtered generaliz...
research
09/01/2017

### A simple en,ex marking rule for degenerate intersection points in 2D polygon clipping

A simple en,ex rule to mark the intersection points of 2D input polygon ...
research
08/30/2019

### On a weighted linear matroid intersection algorithm by deg-det computation

In this paper, we address the weighted linear matroid intersection probl...
research
09/15/2017

### The Dominating Set Problem in Geometric Intersection Graphs

We study the parameterized complexity of dominating sets in geometric in...
research
06/11/2021

### Nawrotzki's Algorithm for the Countable Splitting Lemma, Constructively

We reprove the countable splitting lemma by adapting Nawrotzki's algorit...
research
04/15/2018

### Approximating the covariance ellipsoid

We explore ways in which the covariance ellipsoid B={v ∈R^d : E <X,v>^2...