Iterative building of Barabanov norms and computation of the jointspectral radius for matrix sets

The problem of construction of Barabanov norms for analysis ofproperties of the joint (generalized) spectral radius ofmatrix sets has been discussed in a number of publications. In[18, 21] the method of Barabanovnorms was the key instrument in disproving the Lagarias-WangFiniteness Conjecture. The related constructions wereessentially based on the study of the geometrical properties ofthe unit balls of some specific Barabanov norms. In thiscontext the situation when one fails to find among currentpublications any detailed analysis of the geometricalproperties of the unit balls of Barabanov norms looks a bitparadoxical. Partially this is explained by the fact thatBarabanov norms are defined nonconstructively, by an implicitprocedure. So, even in simplest cases it is very difficult tovisualize the shape of their unit balls. The present work maybe treated as the first step to make up this deficiency. In thepaper an iteration procedure is considered that allows to buildnumerically Barabanov norms for the irreducible matrix sets andsimultaneously to compute the joint spectral radius of thesesets.