以同倫連續法求聯立多項式方程系統的所有孤立解之理論已被 Garcia, Zangwill 和 Drexler 提出，其中最典型的是全次數同倫連續法。追蹤同倫曲線分為預測與校正兩部分，
本論文我們比較不同預測方法在全次數同倫連續法中的效能，包括 Runge-Kutta method, Adams-Bashforth method and cubic Hermite method。此外，我們設計了一套適性調整步長的演算法，此演算法是依據牛頓法校正過程中所提供的資訊，從實驗數據上發現此方法兼顧效率與穩定性。
最後將它運用在解隨機乘積同倫法的特徵值問題上

The theory of solving polynomial systems by homotopy continuation method has been proposed by Garcia, Zangwill and Drexler, and the most typical method in this category is total degree homotpy. The numerical implementation of tracking homotopy curves can be taken as two parts: prediction and correction. In this thesis we compare the performance of several prediction methods in the total degree homotopy, including Runge-Kutta method, Adams-Bashforth method and cubic Hermite method. In addition, we design an adaptive stepsize control algorithm in path tracking, which is based on the information obtained during Newton correction process. The numerical experiment shows that the stepsize control algorithm is quite efficient and reliable in path tracking. In the end we employ the algorithm for solving eigenvalue problems by random product homotopy method

摘要i
Abstract ii
1 Introduction 1
2 Preliminary 4
2.1 Continuationmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Predictionmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Runge-Kuttamethod . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 cubic Hermite interpolation method . . . . . . . . . . . . . . . 7
2.2.3 Adams-Bashforthmethod . . . . . . . . . . . . . . . . . . . . 8
2.3 Newtonmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Numerical experiments of Fixed Stepsize 12
4 Adaptive stepsize control 26
5 Numerical Results of adaptive stepsize control 31
5.1 Polynomial equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Deficient polynomial equations . . . . . . . . . . . . . . . . . . . . . . 39
6 Applications 41
7 Conclusion 43
References 44
A Appendix 46
A.1 The comparison of mathematical software for solving polynomial equations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.2 Solving polynomial equations bymanual . . . . . . . . . . . . . . . . 47
A.3 The local truncation error of numericalmethods . . . . . . . . . . . . 54

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