The level set method for the two-sided eigenproblem
Abstract
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.
Cite
@article{arxiv.1006.5702,
title = {The level set method for the two-sided eigenproblem},
author = {Stephane Gaubert and Sergei Sergeev},
journal= {arXiv preprint arXiv:1006.5702},
year = {2014}
}
Comments
34 pages, 4 figures. Changes with respect to the previous version: we explain relation to mean-payoff games and discrete event systems, and show that the reconstruction of spectrum is pseudopolynomial