Gripenberg-like algorithm for the lower spectral radius

TL;DR

This paper extends Gripenberg's algorithm to compute the lower spectral radius of non-negative matrix sets, providing a rapid convergence method crucial for analyzing minimal growth rates in discrete dynamical systems.

Contribution

The work introduces an extended algorithm employing a time-varying antinorm for the lower spectral radius, building upon and improving Gripenberg's approach for non-negative matrices.

Findings

Algorithm demonstrates rapid convergence.

Effective for non-negative matrix families.

Improves bounds for spectral radius approximation.

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.