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.

翻译：本文提出了一种扩展算法，用于计算有限非负矩阵集的下谱半径。给定矩阵集$\mathcal{F} = \{A_1, \ldots, A_m\}$，下谱半径表示由$\mathcal{F}$生成的乘积半群中序列的最小增长率。该量对于刻画形式为$x_{k+1} = A_{i_k} x_k$（其中对所有$k \ge 0$有$A_{i_k} \in \mathcal{F}$）的离散动力系统的最优稳定轨迹至关重要。对于著名的联合谱半径（表示最高增长率），Gripenberg于1996年提出了一种经典算法，该算法能提供合适的上下界，并能以任意精度逼近联合谱半径。对于下谱半径而言，由于无法直接获得下界（与联合谱半径相反），其计算显得更具挑战性。我们的工作将Gripenberg的方法推广到非负矩阵族的下谱半径计算。所提出的算法采用时变反范数，并展现出快速收敛性。其成功实施与下谱半径可作为Gelfand极限获取的性质相关，该性质最近由Guglielmi和Zennaro（2020）证明。此外，我们还提出了一种改进经典Gripenberg算法的方法，用于逼近任意矩阵集的联合谱半径。