On the accuracy of the ellipsoid norm approximation of the joint spectral radius

Blondel, Vincent [UCL] Nesterov, Yurii [UCL] Theys, Jacques [UCL]

The joint spectral radius of a set of matrices is a measure of the maximal asymptotic growth rate that can be obtained by forming long products of matrices taken from the set. This quantity appears in a number of application contexts but is notoriously difficult to compute and to approximate. We introduce in this paper an approximation \hat \rho that is based on ellipsoid norms, that can be computed by convex optimization, and that is such that the joint spectral radius belongs to the interval [\hat \rho/\sqrt{n},\hat \rho] where n is the dimension of the matrices. We also provide a simple approximation for the special case where the entries of the matrices are non-negative; in this case the approximation is proved to be within a factor at most m (m is the number of matrices) of the exact value.