Fast Methods for Computing the $p$-Radius of Matrices

Abstract

The $p$-radius characterizes the average rate of growth of norms of matrices in a multiplicative semigroup. This quantity has found several applications in recent years. We raise the question of its computability. We prove that the complexity of its approximation increases exponentially with $p$. We then describe a series of approximations that converge to the $p$-radius with a priori computable accuracy. For nonnegative matrices, this gives efficient approximation schemes for the $p$-radius computation.