  When electron-electron correlations are important, it is often necessary to
use exact numerical methods, such as Lanczos diagonalization, to study the full
many-body Hamiltonian. Unfortunately, such exact diagonalization methods are
restricted to small system sizes. We show that if the Hubbard $U$ term is
replaced by a ``periodic Hubbard" term, the full many body Hamiltonian may be
numerically exactly solved, even for very large systems (even $>$100 sites),
though only for low fillings. However, for half-filled systems and large $U$
this approach is not only no longer exact, it no longer improves extrapolation
to larger systems. We discuss how generalized ``randomized variable averaging''
(RVA) or ``phase randomization'' schemes can be reliably employed to improve
extrapolation to large system sizes in this regime. This general approach can
be combined with any many-body method and is thus of broad interest and
applicability.
