Year: 2024
Type: preprint
Abstract: This article presents an extended algorithm for computing the lower spectral radius of finite, non-negative matrix sets. Given a set of matrices $\mathcal{F} = \{A_1, \ldots, A_m\}$, the lower spectral radius represents the minimal growth rate of sequences in the product semigroup generated by $\mathcal{F}$. This quantity is crucial for characterizing optimal stable trajectories in discrete dynamical systems of the form $x_{k+1} = A_{i_k} x_k$, where $A_{i_k} \in \mathcal{F}$ for all $k \ge 0$. For the well-known joint spectral radius (which represents the highest growth rate), a famous algorithm providing suitable lower and upper bounds and able to approximate the joint spectral radius with arbitrary accuracy was proposed by Gripenberg in 1996. For the lower spectral radius, where a lower bound is not directly available (contrarily to the joint spectral radius), this computation appears more challenging. Our work extends Gripenberg's approach to the lower spectral radius computation for non-negative matrix families. The proposed algorithm employs a time-varying antinorm and demonstrates rapid convergence. Its success is related to the property that the lower spectral radius can be obtained as a Gelfand limit, which was recently proved in Guglielmi and Zennaro (2020). Additionally, we propose an improvement to the classical Gripenberg algorithm for approximating the joint spectral radius of arbitrary matrix sets.
Source: arXiv (Cornell University)
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Awards: G5245466801, G598502428
OpenAlex ID: W4403784483