  We present a new approach for constructing polytope Lyapunov functions for
continuous-time linear switching systems (LSS). This allows us to decide the
stability of LSS and to compute the Lyapunov exponent with a good precision in
relatively high dimensions. The same technique is also extended for
stabilizability of positive systems by evaluating a polytope concave Lyapunov
function ("antinorm") in the cone. The method is based on a suitable
discretization of the underlying continuous system and provides both a lower
and an upper bound for the Lyapunov exponent. The absolute error in the
Lyapunov exponent computation is estimated from above and proved to be linear
in the dwell time. The practical efficiency of the new method is demonstrated
in several examples and in the list of numerical experiments with randomly
generated matrices of dimensions up to $10$ (for general linear systems) and up
to $100$ (for positive systems). The development of the method is based on
several theoretical results proved in the paper: the existence of monotone
invariant norms and antinorms for positively irreducible systems, the
equivalence of all contractive norms for stable systems and the linear
convergence theorem.
