[Paper Review] Max-relaxation iteration procedure for building of Barabanov norms: convergence and examples

Victor Kozyakin|arXiv (Cornell University)|Feb 17, 2010

Matrix Theory and Algorithms4 citations

TL;DR

This paper proposes a max-relaxation iteration procedure to numerically construct Barabanov norms for irreducible matrix sets and compute their joint spectral radius with guaranteed convergence and a posteriori error bounds. The method iteratively refines a norm estimate using matrix actions in polar coordinates, ensuring convergence to a Barabanov norm and providing computable bounds on the joint spectral radius.

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.

Motivation & Objective

• To address the lack of constructive methods for visualizing and computing Barabanov norms, which are defined implicitly and nonconstructively.
• To develop a numerical algorithm that builds Barabanov norms for irreducible matrix sets and computes the joint spectral radius simultaneously.
• To provide a posteriori error bounds for the approximation of the joint spectral radius using the iterative procedure.
• To overcome the non-uniqueness and non-convergence issues of direct iteration schemes by introducing a relaxation mechanism.
• To lay the foundation for geometric analysis of Barabanov norm unit balls through numerical computation.

Proposed method

• The algorithm uses a max-relaxation iteration: at each step, the norm is updated via $ \|x\|_{n+1} = \gamma_n^{-1} \max_i \|A_i x\|_n $, where $ \gamma_n $ is a normalization factor.
• The norm is represented on a discrete grid of angles $ \phi \in [0, 2\pi) $, with matrix actions mapped to angle and radius transformations.
• For each direction $ \phi $, the images under $ A_i $ are computed and mapped to the nearest grid point to enable discrete iteration.
• Local convexity is enforced at each step to suppress numerical oscillations and improve stability.
• The joint spectral radius is estimated via $ \rho_n^\pm = \min(R_n / R) $ and $ \max(R_n / R) $, providing bounds on the true value.
• Normalization ensures the unit ball remains bounded and the iteration converges to a Barabanov norm.

Experimental results

Research questions

• RQ1 Can a numerical algorithm be constructed to compute Barabanov norms for irreducible matrix sets with guaranteed convergence?
• RQ2 How can a posteriori error bounds for the joint spectral radius be derived from the iterative process?
• RQ3 Why do direct iteration schemes fail to converge, and can relaxation improve stability?
• RQ4 What is the rate of convergence of the iterative sequence to the true joint spectral radius?
• RQ5 How do the geometric properties of the Barabanov norm unit ball emerge from the iterative construction?

Key findings

• The max-relaxation algorithm converges to a Barabanov norm for any irreducible matrix set, providing a constructive method for an otherwise implicit object.
• The algorithm yields computable a posteriori bounds $ \rho_n^- \leq \rho(\mathscr{A}) \leq \rho_n^+ $, with the interval width decreasing as iterations proceed.
• Numerical experiments show that the algorithm stabilizes and produces visually coherent unit balls for the Barabanov norm, even in non-trivial cases.
• The method successfully handles matrix sets of arbitrary size and number of matrices, with no restriction on the matrix dimension or cardinality.
• The relaxation step is essential: direct iteration without relaxation may fail to converge, highlighting the importance of the normalization and convexity enforcement.
• The algorithm's computational cost depends on the number of grid points, matrix size $ m $, and the choice of averaging function $ \gamma(t,s) $, though no explicit cost estimate is derived.