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2014.7: Max-Plus Singular Values

2014.7: James Hook (2014) Max-Plus Singular Values.

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Abstract

In this paper we prove a new characterization of the max-plus singular values of a max- plus 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.

Item Type:MIMS Preprint
Uncontrolled Keywords:max-plus algebra, singular values, Hungarian scaling, optimal assignment, maximal matching, network flow algorithm, eigenvalues, matrix condition number
Subjects:MSC 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 41 Approximations and expansions
MSC 2000 > 65 Numerical analysis
MSC 2000 > 90 Operations research, mathematical programming
MIMS number:2014.7
Deposited By:Mr James Hook
Deposited On:12 March 2014

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