The level set method for the two-sided eigenproblem
arXiv:1006.5702 · doi:10.1007/s10626-012-0137-z
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.
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
References in corpus (7)
- Tropical linear-fractional programming and parametric mean payoff games
- Best approximation in max-plus semimodules
- The tropical analogue of polar cones
- Tropical polar cones, hypergraph transversals, and mean payoff games
- Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem
- Perturbation of eigenvalues of matrix pencils and optimal assignment problem
- Cyclic projectors and separation theorems in idempotent convex geometry