  When a quantum many-body system undergoes a quench, the time-averaged
density-matrix $\rho$ governs the time-averaged expectation value of any
observable. It is therefore the key object to look at when comparing results
with equilibrium predictions. We show that the weights of $\rho$ can be
efficiently computed with Lanczos diagonalization for relatively large Hilbert
spaces. As an application, we investigate the crossover from perturbative to
non-perturbative quenches in the non-integrable Bose-Hubbard model: on finite
systems, an approximate Boltzmann distribution is observed for small quenches,
while for larger ones the distributions do not follow standard equilibrium
predictions. Studying thermodynamical features, such as the energy fluctuations
and the entropy, shows that $\rho$ bears a memory of the initial state.
