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 singu- lar values quickly and accurately. As well as capturing the asymptotic behavior of the singular values of classical matrices whose entries are exponentially pa- rameterized we show experimentally that max-plus singular values give order of magnitude approximations to the classical singular values of parameter inde- pendent 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 2-norm condition number and that this action can be explained using our new theory for max-plus singular values. Keywords: tropical algebra, max-plus algebra, singular values, diagonal scaling, condition number, optimal assignment problem 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: | 21 Sep 2015 |
| Last Modified: | 20 Oct 2017 14:13 |
| URI: | https://eprints.maths.manchester.ac.uk/id/eprint/2384 |
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Max-Plus Singular Values. (deposited 12 Mar 2014)
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