以同倫連續法求解多項式系統會面臨的一個問題是，若初始系統的解遠多於目標系統的解個數，則當同倫參數接近終點時，有很多曲線會發散。跟蹤發散的曲線不會得到目標系統的解且其計算量也大很多。如何在同倫參數接近終點時，有效的判斷一個曲線是否發散以減少跟蹤發散曲線的計算量稱為末局問題。
本文章首先對 Morgan et al. 所提出的曲線表示式理論來處理末局問題，此理論說對於以同倫連續法解多項式系統的同倫曲線，當參數接近終點時，曲線可以被皮瑟級數所展開；更有甚者，皮瑟級數的首項指數可以決定曲線的收斂性。接著，我們將代數特徵值問題轉換成求解多項式系統問題，再以同倫連續法解之，探討對於非退化代數特徵值問題的同倫曲線性質與其面臨的末局問題。

The homotopy continuation method is considered to solve polynomial systems. If the number of solutions of the starting system is much more than that of target system, many of curves will diverge when the homotopy parameter goes to the end. In this case we will have a difficulty in tracing the solution curves by continuation method because tracing divergent curves does not help us obtain any solution of target system but also its computation is very costly. As the parameter goes to the end, how to determine whether a curve converges or diverges effectively for reducing the computational cost is called the end game problem.
In this thesis we will deal with the end game problem via the curve expression theory proposed by Morgan et al. The theory says the homotopy curve for solving polynomial systems can be expressed by Puiseux series expansion when the parameter is nearby the end. Moreover, the exponents of the leading term in Puiseux series determine the convergency of a curve. We also study the algebraic eigenvalue problems solving by homotopy continuation method. Several observations in numerical experiments for nonderogatory algebraic eigenvalue problem and its end game problem will be reported.

目錄
1 背景簡介. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 初步工具與理論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 全次數同倫. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 投影同倫. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3
2.3 投影牛頓迭代法. . . . . . . . . . . . . . . . . . . . . . . . . . . .6
2.4 Jordan 典型形式. . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 皮瑟級數展開式解決末局問題的應用. . . . . . . . . . . . . . . 10
3.1 皮瑟級數展開式. . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
3.2 估計曲線方向. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 提高曲線方向的精確度. . . . . . . . . . . . . . . . . . . . . . . . 14
4 Cyclic-5 的數值結果. . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1 Cyclic-5 的循環解. . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 全次數同倫解Cyclic-5 的數值結果. . . . . . . . . . . . . . . . 19
4.3 投影同倫解Cyclic-5 的數值結果. . . . . . . . . . . . . . . . . .20
4.4 皮瑟級數展開式解Cyclic-5 的數值結果. . . . . . . . . . . . .21
5 退化及非退化特徵值問題及其數值結果. . . . . . . . . . . . . . 22
6 討論與結語. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
7 附錄. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
8 參考文獻. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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