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.
This is a benchmark set of matrices that has been studied in [10] , REF mainly because it provides a "worst-case" example for the method of common quadratic Lyapunov functions.
Approximation of the joint spectral radius using sum of squares
Linear Algebra and its Applications, Vol. 428, No. 10, pp. 2385-2402, 2008.
Mathematics