Stability of linear problems: joint spectral radius of sets of matrices

It is well known that the stability analysis of step-by-step numerical methods for differential equations often reduces to the analysis of linear difference equations with variable coefficients. This class of difference equations leads to a family $\mathcal{F}$ of matrices depending on some parameters and the behaviour of the solutions depends on the convergence properties of the products of the matrices of $\mathcal{F}$. To date, the techniques mainly used in the literature are confined to the search for a suitable norm and for conditions on the parameters such that the matrices of $\mathcal{F}$ are contractive in that norm. In general, the resulting conditions are more restrictive than necessary. An alternative and more effective approach is based on the concept of {\em joint spectral radius} of the family $\mathcal{F}$, $\rho(\mathcal{F})$. It is known that all the products of matrices of $\mathcal{F}$ asymptotically vanish if and only if $\rho(\mathcal{F})<1$. The aim of this chapter is that to discuss the main theoretical and computational aspects involved in the analysis of the joint spectral radius and in applying this tool to the stability analysis of the discretizations of differential equations as well as to other stability problems. In particular, in the last section, we present some recent euristic techniques for the search of optimal products in finite families, which constitutes a fundamental step in the algorithms which we discuss. The material we present in the final section is part of an original research which is in progress and is still unpublished