An Efficient Algorithm for Nontrivial Eigenvectors in Max-Plus Algebra

Mubasher Umer,Umar Hayat,Fazal Abbas

doi:10.3390/sym11060738

Abstract

The eigenproblem for matrices in max-plus algebra describes the steady state of the system, and therefore it has been intensively studied by many authors. In this paper, we propose an algorithm to compute the eigenvalue and the corresponding eigenvectors of a square matrix in an iterative way. The algorithm is extended to compute the nontrivial eigenvectors for Latin squares in max-plus algebra.

Highlights

Discrete event systems can be used to study processes that are driven by the occurrence of events.The relevant variables represent times at which events are taking place
An important class of systems consists of min-max-plus systems in which the operations—the maximization, the minimization and the addition, can describe discrete event dynamic systems
The union of two sets of equations describes the min-max-plus systems, where the maximization and addition are contained in one set of equations and the minimization and addition are contained in another set of equations

Summary

Introduction

Discrete event systems can be used to study processes that are driven by the occurrence of events. Max-plus algebra Re = R ∪ {e} where e = −∞, is a set which deals with two main operations—⊕ as maximization and ⊗ as addition. These operations endow Re with the structure of a commutative semiring. A novel iterative method is proposed for the computation of eigenvalue and the corresponding eigenvectors in max-plus algebra. A new algorithm was proposed to solve systems of max-plus linear inequalities [13]. This algorithm uses the technique of cyclic projectors. A new algorithm is proposed to determine the eigenvalue and eigenvectors in an iterative way.

Max-Plus Algebra

Latin Square

Conclusions