  We develop a new approach to the computation of the Hausdorff dimension of
the invariant set of an iterated function system or IFS. In the one dimensional
case, our methods require only C^3 regularity of the maps in the IFS. The key
idea, which has been known in varying degrees of generality for many years, is
to associate to the IFS a parametrized family of positive, linear,
Perron-Frobenius operators L_s. The operators L_s can typically be studied in
many different Banach spaces. Here, unlike most of the literature, we study L_s
in a Banach space of real-valued, C^k functions, k >= 2; and we note that L_s
is not compact, but has a strictly positive eigenfunction v_s with positive
eigenvalue lambda_s equal to the spectral radius of L_s. Under appropriate
assumptions on the IFS, the Hausdorff dimension of the invariant set of the IFS
is the value s=s_* for which lambda_s =1. This eigenvalue problem is then
approximated by a collocation method using continuous piecewise linear
functions (in one dimension) or bilinear functions (in two dimensions). Using
the theory of positive linear operators and explicit a priori bounds on the
derivatives of the strictly positive eigenfunction v_s, we give rigorous upper
and lower bounds for the Hausdorff dimension s_*, and these bounds converge to
s_* as the mesh size approaches zero.
