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

Universal algorithms for computing spectra of periodic operators