On tropical linear systems

• Main Authors: You, Jiun Bo, 游竣博
• Other Authors: Tsai, Yen Lung
• Format: Others
• Language: zh-TW
• Online Access: http://ndltd.ncl.edu.tw/handle/62487474769917780698

碩士 === 國立政治大學 === 應用數學研究所 === 100 === The thesis mainly discusses the methods of finding solutions of tropical linear systems A x = b and two-sided homogeneous tropical linear systems A x = B y. We are able to give explicit descriptions of all solutions of any tropical linear systems A x = b and two-sided homogeneous tropical linear systems A x = B y. As the classical situations, when solving the linear systems of the form A x = b, we first find the solutions for the corresponding ``homogeneous'' case A x = 0. For two-sided homogeneous tropical linear systems A x = B y, we use the concept of win sequence to convert it into a finite number k of classical linear systems: either a system S: C[x^t -y^t 1]^t = 0 of equations or a system T: D[x^t -y^t 1]^t <= 0 of inequalities. Moreover, we used so called ``compatibility conditions'' to reduce the number of k. The particular feature of both S and T is that each item (equation or inequality) is bivariate. It involves exactly two variables; one variable with coefficient 1, and the other one with -1. S is solved by Gauss-Jordon elimination. We explain how to solve T by a method similar to Gauss-Jordon elimination. To achieve this, we introduce the notion of sub–special matrix. The procedure applied to T is called sub–specialization. Finally, we will use MATLAB to solve tropical linear systems of these two types.