The level set method for the two-sided eigenproblem

Details The level set method for the two-sided eigenproblem The level set method for the two-sided eigenproblem

2010-06-29

arxiv.org/abs/1006.5702v2

Mathematics

Metric Geometry

34 pages, 4 figures. Changes with respect to the previous version: we explain relation to mean-payoff games and discrete event

Scientific paper

We consider the max-plus analogue of the eigenproblem for matrix pencils Ax=lambda Bx. We show that the spectrum of (A,B) (i.e., the set of possible values of lambda), which is a finite union of intervals, can be computed in pseudo-polynomial number of operations, by a (pseudo-polynomial) number of calls to an oracle that computes the value of a mean payoff game. The proof relies on the introduction of a spectral function, which we interpret in terms of the least Chebyshev distance between Ax and lambda Bx. The spectrum is obtained as the zero level set of this function.

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