  We show that the joint spectral radius of a finite collection of nonnegative
matrices can be bounded by the eigenvalue of a non-linear operator. This
eigenvalue coincides with the ergodic constant of a risk-sensitive control
problem, or of an entropy game, in which the state space consists of all
switching sequences of a given length. We show that, by increasing this length,
we arrive at a convergent approximation scheme to compute the joint spectral
radius. The complexity of this method is exponential in the length of the
switching sequences, but it is quite insensitive to the size of the matrices,
allowing us to solve very large scale instances (several matrices in dimensions
of order 1000 within a minute). An idea of this method is to replace a
hierarchy of optimization problems, introduced by Ahmadi, Jungers, Parrilo and
Roozbehani, by a hierarchy of nonlinear eigenproblems. To solve the latter
eigenproblems, we introduce a projective version of Krasnoselskii-Mann
iteration. This method is of independent interest as it applies more generally
to the nonlinear eigenproblem for a monotone positively homogeneous map. Here,
this method allows for scalability by avoiding the recourse to linear or
semidefinite programming techniques.
