  We prove a universal limit theorem for the halting time, or iteration count,
of the power/inverse power methods and the QR eigenvalue algorithm.
Specifically, we analyze the required number of iterations to compute extreme
eigenvalues of random, positive-definite sample covariance matrices to within a
prescribed tolerance. The universality theorem provides a complexity estimate
for the algorithms which, in this random setting, holds with high probability.
The method of proof relies on recent results on the statistics of the
eigenvalues and eigenvectors of random sample covariance matrices (i.e.,
delocalization, rigidity and edge universality).
