Max-Plus Singular Values

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

This is the latest version of this item.

[img] PDF
MAX-PLUS-SV.pdf

Download (515kB)

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: http://eprints.maths.manchester.ac.uk/id/eprint/2384

Available Versions of this Item

Actions (login required)

View Item View Item