Approximation of the joint spectral radius using sum of squares

TL;DR

This paper introduces a computationally efficient SOS-based method to approximate the joint spectral radius of matrices, providing tighter bounds and unifying previous results.

Contribution

It presents a novel SOS programming approach for joint spectral radius approximation with proven asymptotic tightness and independence from matrix set size.

Findings

Provides a bound on approximation quality that unifies earlier results.

Demonstrates tighter approximations compared to previous methods.

Includes numerical experiments validating the approach.

Abstract

We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices. Additionally, we present a comparison between our approximation scheme and earlier techniques, including the use of common quadratic Lyapunov functions and a method based on matrix liftings. Theoretical results and numerical investigations show that our approach yields tighter approximations.