In this paper the problem of the computation of the joint spectral radius of a finite set of matrices is considered. We present an algorithm which, under some suitable assumptions, is able to check if a certain product in the multiplicative semigroup is spectrum maximizing. The algorithm proceeds by attempting to construct a suitable extremal norm for the family, namely a complex polytope norm. As examples for testing 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 arising in the zero-stability analysis of the 4-step BDF formula for ordinary differential equations.
