  Schr\"odinger operators with periodic (possibly complex-valued) potentials
and discrete periodic operators (possibly with complex-valued entries) are
considered, and in both cases the computational spectral problem is
investigated: namely, under what conditions can a "one-size-fits-all" algorithm
for computing their spectra be devised? It is shown that for periodic banded
matrices this can be done, as well as for Schr\"odinger operators with periodic
potentials that are sufficiently smooth. In both cases implementable algorithms
are provided, along with examples. For certain Schr\"odinger operators whose
potentials may diverge at a single point (but are otherwise well-behaved) it is
shown that there does not exist such an algorithm, though it is shown that the
computation is possible if one allows for two successive limits.
