An algorithm for finding extremal polytope norms of matrix families

In this paper the problem of the computation of the joint spectral radius of a finite set of matrices isconsidered. We present an algorithm which, under some suitable assumptions, is able to check if a certainproduct in the multiplicative semigroup is spectrum maximizing. The algorithm proceeds by attemptingto construct a suitable extremal norm for the family, namely a complex polytope norm. As examples fortesting our technique, we first consider the set of two 2-dimensional matrices recently analyzed by Blondel,Nesterov and Theys to disprove the finiteness conjecture, and then a set of 3-dimensional matrices arisingin the zero-stability analysis of the 4-step BDF formula for ordinary differential equations.