  We address the problem of the exact computation of two joint spectral
characteristics of a family of linear operators, the joint spectral radius (in
short JSR) and the lower spectral radius (in short LSR), which are well-known
different generalizations to a set of operators of the usual spectral radius of
a linear operator. In this article we develop a method which - under suitable
assumptions - allows to compute the JSR and the LSR of a finite family of
matrices exactly. We remark that so far no algorithm was available in the
literature to compute the LSR exactly.
  The paper presents necessary theoretical results on extremal norms (and on
extremal antinorms) of linear operators, which constitute the basic tools of
our procedures, and a detailed description of the corresponding algorithms for
the computation of the JSR and LSR (the last one restricted to families sharing
an invariant cone). The algorithms are easily implemented and their
descriptions are short.
  If the algorithms terminate in finite time, then they construct an extremal
norm (in the JSR case) or antinorm (in the LSR case) and find their exact
values; otherwise they provide upper and lower bounds that both converge to the
exact values. A theoretical criterion for termination in finite time is also
derived. According to numerical experiments, the algorithm for the JSR finds
the exact value for the vast majority of matrix families in dimensions less
than 20. For nonnegative matrices it works faster and finds JSR in dimensions
of order 100 within a few iterations; the same is observed for the algorithm
computing the LSR. To illustrate the efficiency of the new method we are able
to apply it in order to give answers to several conjectures which have been
recently stated in combinatorics, number theory, and the theory of formal
languages.
