Max-relaxation iteration procedure for building of Barabanov norms: convergence and examples. (arXiv:1002.3251v1 [math.RA])

Max-relaxation iteration procedure for building of Barabanov norms: convergence and examples. (arXiv:1002.3251v1 [math.RA]): "

The problem of construction of Barabanov norms for analysis of properties of
the joint (generalized) spectral radius of matrix sets has been discussed in a
number of publications. In previous papers of the author the method of
Barabanov norms was the key instrument in disproving the Lagarias-Wang
Finiteness Conjecture. The related constructions were essentially based on the
study of the geometrical properties of the unit balls of some specific
Barabanov norms. In this context the situation when one fails to find among
current publications any detailed analysis of the geometrical properties of the
unit balls of Barabanov norms looks a bit paradoxical. Partially this is
explained by the fact that Barabanov norms are defined nonconstructively, by an
implicit procedure. So, even in simplest cases it is very difficult to
visualize the shape of their unit balls. The present work may be treated as the
first step to make up this deficiency. In the paper an iteration procedure is
considered that allows to build numerically Barabanov norms for the irreducible
matrix sets and simultaneously to compute the joint spectral radius of these
sets.

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