  We determine the probability, structure dependent, that the block Wiedemann
algorithm correctly computes leading invariant factors. This leads to a tight
lower bound for the probability, structure independent. We show, using block
size slightly larger than $r$, that the leading $r$ invariant factors are
computed correctly with high probability over any field. Moreover, an algorithm
is provided to compute the probability bound for a given matrix size and thus
to select the block size needed to obtain the desired probability. The worst
case probability bound is improved, post hoc, by incorporating the partial
information about the invariant factors.
