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On the maximal solution of a linear system over tropical semirings
Mathematical Sciences ( IF 2 ) Pub Date : 2020-05-05 , DOI: 10.1007/s40096-020-00325-w
Sedighe Jamshidvand , Shaban Ghalandarzadeh , Amirhossein Amiraslani , Fateme Olia

Nowadays, certain problems in automata theory, manufacturing systems, telecommunication networks, parallel processing systems and traffic control are intimately linked with linear systems over tropical semirings. Due to non-invertibility of matrices—except monomial matrices—over certain semirings, we cannot generally take advantage of having the inverse of the coefficient matrix of a system to solve it. The main purpose of this paper is to introduce two methods based on the pseudo-inverse of a matrix for solving a linear system of equations over tropical semirings. To this end, under suitable conditions, we first reduce the order of the system through some row–column operational analysis. We then present a necessary and sufficient condition for the system to have a maximal solution. This solution is also obtained through a new version of Cramer’s rule for overdetermined system of equations. Finally, some illustrative examples are given to show the efficiency of the proposed methods, and Maple procedures are also included in the end.

中文翻译:

关于热带半环上线性系统的最大解

如今,自动机理论,制造系统,电信网络,并行处理系统和交通控制中的某些问题已与热带半环上的线性系统紧密联系在一起。由于矩阵在某些半环上的不可逆性(单项矩阵除外),我们通常无法利用使系统的系数矩阵逆来求解的优势。本文的主要目的是介绍两种基于矩阵拟逆的方法,用于求解热带半环上的线性方程组。为此,在适当的条件下,我们首先通过一些行-列操作分析来减少系统的顺序。然后,我们提出了系统具有最大解的必要和充分条件。该解决方案还可以通过新版本的Cramer规则(针对超定方程组)获得。最后,给出了一些说明性的例子来说明所提出方法的效率,最后还包括了Maple程序。
更新日期:2020-05-05
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