  We introduce a new approach to evaluate the largest Lyapunov exponent of a
family of nonnegative matrices. The method is based on using special positive
homogeneous functionals on $R^{d}_+,$ which gives iterative lower and upper
bounds for the Lyapunov exponent. They improve previously known bounds and
converge to the real value. The rate of convergence is estimated and the
efficiency of the algorithm is demonstrated on several problems from
applications (in functional analysis, combinatorics, and lan- guage theory) and
on numerical examples with randomly generated matrices. The method computes the
Lyapunov exponent with a prescribed accuracy in relatively high dimensions (up
to 60). We generalize this approach to all matrices, not necessar- ily
nonnegative, derive a new universal upper bound for the Lyapunov exponent, and
show that such a lower bound, in general, does not exist.
