A relaxation scheme for computation of the joint spectral radius of matrix sets

Abstract

The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In this 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.

Keywords:

• infinite matrix products
• generalized spectral radius
• joint spectral radius
• extremal norms
• Barabanov norms
• irreducibility
• numerical algorithms

AMS Subject Classification::

• 15A18
• 15A60
• 65F15

Acknowledgements

This work was supported by the Russian Foundation for Basic Research, projects nos. 06-01-00256, 09-01-00119. The author is grateful to the reviewer for a number of valuable remarks.

Notes

1. A matrix set A is called irreducible, if the matrices from A have no common invariant subspaces except {0} and . In 19-21 such a matrix set was called quasi-controllable.

2. The set is called body if it contains at least one interior point.