本篇論文主要在探討熱帶線性系統(tropical linear system) A x = b 與雙邊齊次熱帶線性系統(two-sided homogeneous tropical linear system) A x = B y 的求解方法。我們將明確的描述任何熱帶線性系統與雙邊齊次熱帶線性系統的解。

如同古典的論述, 當求解線性系統 A x = b 時, 我們首先會先找到對應的 ``齊次'' 系統 A x = 0 來求解。而對於雙邊齊次熱帶線性系統, 我們將利用勝序列的概念, 將雙邊齊次熱帶線性系統轉化為 k 組古典熱帶線性系統: 含等式系統 S: C[x^t -y^t 1]^t = 0 與不等式系統 T: D[x^t -y^t 1]^t <= 0 。除此之外, 利用相容性條件來減少 k 的數量。

過程中我們處理的 S, T 均為雙變量的系統, 係數分別為 1 與 -1, 對於 S 我們以高斯-喬登消去法(Gauss–Jordan elimination)處理。對於 T 我們將以類似高斯-喬登消去法的方式進行列運算, 因此我們定義次特殊矩陣(sub-special matrix), 而進行的過程我們稱之為次特殊化(sub–specialization)。

最後將以 MATLAB 作為工具來求解出這兩類的熱帶線性系統。

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.

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