  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.
