Max-Plus Singular Values

Hook, James (2014) Max-Plus Singular Values. [MIMS Preprint]

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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 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 41 Approximations and expansions MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis MSC 2010, the AMS's Mathematics Subject Classification > 90 Operations research, mathematical programming
Depositing User:	Mr James Hook
Date Deposited:	12 Mar 2014
Last Modified:	20 Oct 2017 14:13
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2112

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• Max-Plus Singular Values. (deposited 12 Mar 2014) [Currently Displayed]
  • Max-Plus Singular Values. (deposited 21 Sep 2015)

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