A comparison of approximation algorithms for the joint spectral radius

Let Σ be a set of n × n matrices and let us consider the fol-lowing linear iteration: x(t +1) = Atx(t), At ∈ Σ for all t. These kinds of switched linear systems arise in many ap-plications such as asynchronous systems, hybrid systems, switching control,... The stability under arbitrary switchings of this system de-pends on a quantity called the joint spectral radius (JSR), which represents the maximal growth rate of such a discrete linear system. More precisely, the JSR of a set Σ of matrices is defined by the following expression: ρ(Σ) = lim k→∞ max{‖Ai1...Aik‖1/k | Ai ∈ Σ}