Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets
arXiv:0810.2154 · doi:10.3934/dcdsb.2010.14.143
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. 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 two iteration procedure are considered that allow to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.
17 pages, 36 bibliography references, 3 figures; shortened version, new LaTeX style, fixed typos, accepted in DCDS-B
References in corpus (4)
- Approximation of the joint spectral radius using sum of squares
- On accuracy of approximation of the spectral radius by the Gelfand formula
- Estimates of Amplitudes of Transient Regimes in Quasi-Controllable Discrete Systems
- Max-relaxation iteration procedure for building of Barabanov norms: convergence and examples
Cited by in corpus (6)
- Mather sets for sequences of matrices and applications to the study of joint spectral radii
- An explicit Lipschitz constant for the joint spectral radius
- Rank-one Characterization of Joint Spectral Radius of Finite Matrix Family
- Improved invariant polytope algorithm and applications
- On pairs of spectrum maximizing products with distinct factor multiplicities
- Non-Sturmian sequences of matrices providing the maximum growth rate of matrix products