We find analytic upper and lower bounds of the Lyapunov exponents of the
product of random matrices related to the one-dimensional disordered Ising
model, using a deterministic map which transforms the original system into a
new one with smaller average couplings and magnetic fields. The iteration of the
map gives bounds which estimate the Lyapunov exponents with increasing
accuracy. We prove, in fact, that both the upper and the lower bounds converge
to the Lyapunov exponents in the limit of infinite iterations of the map.
A formal expression of the Lyapunov exponents is thus obtained in terms of the
limit of a sequence. Our results allow us to introduce a new numerical procedure
for the computation of the Lyapunov exponents which has a precision higher
than Monte Carlo simulations