  Convergence rates of block iterations for solving eigenvalue problems
typically measure errors of Ritz values approximating eigenvalues. The errors
of the Ritz values are commonly bounded in terms of principal angles between
the initial or iterative subspace and the invariant subspace associated with
the target eigenvalues. Such bounds thus cannot be applied repeatedly as needed
for restarted block eigensolvers, since the left- and right-hand sides of the
bounds use different terms. They must be combined with additional bounds which
could cause an overestimation. Alternative repeatable bounds that are
angle-free and depend only on the errors of the Ritz values have been pioneered
for Hermitian eigenvalue problems in doi:10.1515/rnam.1987.2.5.371 but only for
a single extreme Ritz value. We extend this result to all Ritz values and
achieve robustness for clustered eigenvalues by utilizing nonconsecutive
eigenvalues. Our new bounds cover the restarted block Lanczos method and its
modifications with shift-and-invert and deflation, and are numerically
advantageous.
