Mufid, Muhammad Syifa'ul, Patel, Ebrahim ORCID: https://orcid.org/0000-0001-9473-6542 and Sergeev, Sergeĭ (2023) Solving linear equations over maxmin-ω systems. Linear Algebra and its Applications, 681. pp. 21-46. ISSN 0024-3795 (Print), 1873-1856 (Online) (doi:10.1016/j.laa.2023.10.012)

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Abstract

Maxmin-ω dynamical systems were previously introduced as an “all-in-one package” that can yield a solely min-plus, a solely max-plus, or a max-min-plus dynamical system by varying a parameter . With such systems in mind, it is natural to introduce and consider maxmin-ω linear systems of equations of the type. However, to our knowledge, such maxmin-ω linear systems have not been studied before and in this paper we present an approach to solve them. We show that the problem can be simplified by performing normalization and then generating a “canonical” matrix which we call the principal order matrix. Instead of directly trying to find the solutions, we search the possible solution indices which can be identified using the principal order matrix and the parameter ω. The fully active solutions are then immediately obtained from these solution indices. With the fully active solutions at hand, we then present the method to find other solutions by applying a relaxation, i.e., increasing or decreasing some components of fully active solutions. This approach can be seen as a generalization of an approach that could be applied to solve max-plus or min-plus linear systems. Our results also shed more light on an unusual feature of maxmin-ω linear systems, which, unlike in the usual linear algebra, can have a finite number of solutions in the case where their solution is non-unique.

Item Type:	Article
Uncontrolled Keywords:	min-plus system, max-plus system, maxmin-ω system, linear equation
Subjects:	Q Science > QA Mathematics
Faculty / School / Research Centre / Research Group:	Faculty of Engineering & Science Faculty of Engineering & Science > School of Computing & Mathematical Sciences (CMS)
Last Modified:	20 Sep 2024 10:16
URI:	http://gala.gre.ac.uk/id/eprint/48115

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