Max-plus Singular Values

In this paper we prove a new characterization of the max-plus singular values of a maxplus matrix, as the max-plus eigenvalues of an associated max-plus matrix pencil. This new characterization allows us to compute max-plus singular values quickly and accurately. As well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially parameterized we show experimentally that max-plus singular values give order of magnitude approximations to the classical singular values of parameter independent classical matrices. We also discuss Hungarian scaling, which is a diagonal scaling strategy for preprocessing classical linear systems. We show that Hungarian scaling can dramatically reduce the d-norm condition number and that this action can be explained using our new theory for max-plus singular values. Introduction Max-plus algebra concerns the semiring Rmax = R ∪ {−∞} with addition and multiplication operations a⊕ b = max{a, b}, a⊗ b = a+ b, a, b ∈ Rmax. More generally tropical algebra is the study of any semiring in which the addition operation is max or min, for example max-times, min-max and max-average. Max-plus algebra naturally describes certain dynamical systems and operations research problems [1]. Max-plus algebra can also be used to approximate or bound the solutions to certain classical algebra problems, which is the topic of this paper. An n×m max-plus matrix G ∈ Rn×m max is simply an n×m array of entries from Rmax. The maxplus Singular Value Decomposition (SVD) of a max-plus matrix was introduced by De Schutter and De Moor in [2]. They work in the max-plus algebra of pairs, which is roughly max-plus algebra with subtraction. In this setting equalities are replaced with weaker relations, which they call balances. Their main result is proving the existence of a max-plus SVD which looks exactly as the classical SVD but with max replacing sum, sum replacing times and balancing replacing equality. The max-plus SVD is useful for analyzing certain max-plus linear systems. De Schutter and De Moore also use the decomposition to introduce a definition of the rank of a max-plus matrix, which ∗School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK ([email protected]). This work was supported by Engineering and Physical Sciences Research Council (EPSRC) grant EP/I005293 ’́Nonlinear Eigenvalue Problems: Theory and Numerics”.