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| Title: | 熱帶線性系統之研究
On tropical linear systems |
| Authors: | 游竣博
You, Jiun Bo |
| Contributors: | 蔡炎龍
Tsai, Yen Lung
游竣博
You, Jiun Bo |
| Keywords: | 熱帶線性系統
tropical linear system |
| Date: | 2011 |
| Issue Date: | 2012-04-17 10:25:01 (UTC+8) |
| Abstract: | 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. |
| Reference: | [1] Fran{\\c{c}}ois Baccelli, Guy Cohen, Geert Jan Olsder, and Jean Pierre Quadrat. Synchronization and linearity-an algebra for discrete event systems, 1992. 1
[2] Diane Maclagan and Bernd Sturmfels. Introduction to tropical geometry, Nov 2009. 1
[3] Grigory Mikhalkin. Tropical geometry and its applications, May 2006. 1
[4] J{"{u}}rgen Richter-Gebert, Bernd Sturmfels, and Thorsten Theobald. First steps in tropical geometry, Dec 2003. 1
[5] David Speyer and Bernd Sturmfels. Tropical mathematics, Aug 2004. 1
[6] P. Butkovi{\\v{c}} and R.A. Cuninghame-Green. The equation A x = B y over (max, +). Theoretical Computer Science, 293(1):3-12, Feb 2003. 1, 3, 16, 24
[7] E. Lorenzo and M. J. de la Puente. An algorithm to describe the solution set of any tropical linear system A x = B x, jul 2010. 1, 3
[8] Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence. Linear algebra, Jul 2003. 3
[9] MathWorks. http://www.mathworks.com/products/matlab/. 27
[10] 張智星. Matlab 程式設計與應用, Sep 2000. 27
[11] David Fass. http://www.mathworks.com/matlabcentral/ leexchange/5475-cartprod-cartesian-product-of-multiple-sets, Jul 2004. 40
[12] David Fass. http://www.mathworks.com/matlabcentral/ leexchange/5476-ind2subvect-multiple-subscript-vector-from-linear-index, Jul 2004. 42 |
| Description: | 碩士
國立政治大學
應用數學研究所
98751001
100 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0098751001 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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Files in This Item:
| File |
Size | Format | |
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| 100101.pdf | 893Kb | Adobe PDF2 | 825 | View/Open |
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