  In this paper we obtain the rates of convergence of the algorithms given in
[13] and [14] for an automatic computation of the centered Hausdorff and
packing measures of a totally disconnected self-similar set. We evaluate these
rates empirically through the numerical analysis of three standard classes of
self-similar sets, namely, the families of Cantor type sets in the real line
and the plane and the class of Sierpinski gaskets. For these three classes and
for small contraction ratios, sharp bounds for the exact values of the
corresponding measures are obtained and it is shown how these bounds
automatically yield estimates of the corresponding measures, accurate in some
cases to as many as 14 decimal places. In particular, the algorithms accurately
recover the exact values of the measures in all cases in which these values are
known by geometrical arguments. Positive results, which confirm some
conjectural values given in [13] and [14] for the measures, are also obtained
for an intermediate range of larger contraction ratios. We give an argument
showing that, for this range of contraction ratios, the problem is inherently
computational in the sense that any theoretical proof, such as those mentioned
above, might be impossible, so that in these cases, our method is the only
available approach. For contraction ratios close to those of the connected case
our computational method becomes intractably time consuming, so the computation
of the exact values of the packing and centered Hausdorff measures in the
general case, with the open set condition, remains a challenging problem.
