解非線性方程式雖然有許多數學標準方法，但是在高維度的求解以及有無窮多解的問題上，現有的方法可以計算出來的結果仍然非常有限，我們希望可以提出一個簡單快速的方法，可以了解無窮多解的分布狀況，並且在局部區域也能找出精確解，同時希望對這些解有可視化的了解。我們利用SVM的特性開發了一個新的方法，可以同時達到以上目標。

There are many standard mathematical methods for solving nonlinear equations. But when it comes to equations in high dimension with infinite solutions, the results from current methods are quite limited. We present a simple fast way which could tell the distribution of these infinite solutions and is capable of finding accurate approximations. In the same time, we also want to have a visual understanding about the roots. Using the features of SVM, we have developed a new method that achieves the above goals.

致謝 i
中文摘要 ii
Abstract iii
Contents iv
List of Tables vi
List of Figures vii
1 Introduction 1
1.1 Nonlinear equations of one variable 1
1.1.1 Bisection method 2
1.1.2 False position method (or Regula falsi) 3
1.1.3 Fixedpoint iteration 4
1.1.4 Wegstein’s method 6
1.1.5 Newton’s method 8
1.1.6 Secant method 10
1.1.7 Steffensen’s method 11
1.1.8 Halley’s method 12
1.2 Nonlinear equations of several variables 14
1.2.1 Directional Newton Method 15
1.2.2 Directional Secant Method 15
1.2.3 Broyden Method 16
2 Methodology 18
2.1 Support Vector Machine (SVM) 19
2.2 A New Method 22
3 Results 28
4 Conclusion 38
Bibliography 40

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