  We present the results of an empirical study of the performance of the QR
algorithm (with and without shifts) and the Toda algorithm on random symmetric
matrices. The random matrices are chosen from six ensembles, four of which lie
in the Wigner class. For all three algorithms, we observe a form of
universality for the deflation time statistics for random matrices within the
Wigner class. For these ensembles, the empirical distribution of a normalized
deflation time is found to collapse onto a curve that depends only on the
algorithm, but not on the matrix size or deflation tolerance provided the
matrix size is large enough (see Figure 4, Figure 7 and Figure 10). For the QR
algorithm with the Wilkinson shift, the observed universality is even stronger
and includes certain non-Wigner ensembles. Our experiments also provide a
quantitative statistical picture of the accelerated convergence with shifts.
