Max-relaxation iteration procedure for building of Barabanov norms: convergence and examples

TL;DR

This paper introduces a new iterative method for constructing Barabanov norms, enabling numerical analysis of the joint spectral radius and the geometric properties of these norms' unit balls.

Contribution

The paper presents the first detailed analysis and a practical iteration procedure for constructing Barabanov norms and studying their geometric properties.

Findings

The iteration procedure successfully constructs Barabanov norms for irreducible matrix sets.

The method allows numerical computation of the joint spectral radius.

It provides insights into the geometry of the unit balls of Barabanov norms.

Abstract

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.