就Laplace特徵值問題而言， 本論文提出Trefftz法(即邊界近似法)的新演算法， 用以解決Helmholtz方程，和使用迭代法以產生特徵值和特徵函數的近似解。新的迭代法擁有超線性的收歛速率，及在數值測試中，和其它流行的求根法相較亦有更好的表現。

此外，在一個單位正方形Dirichlet條件，特徵值問題的基本模型上，我們使用片斷特解。而這個使用片斷特解的新演算法，是非常適合尋找特徵值問題的高精度解， 特別是在那些有多個奇異點， 界面和無界的區域上。因為整個求解域的一致性特解並不總是存在，所以使用片斷特解，同時也有在求解複雜的問題上的好處。

For Laplace's eigenvalue problems, this thesis presents new algorithms of the Trefftz method (i.e. the boundary approximation method), which solve the Helmholtz equation and then use a iteration process to yield approximate eigenvalues and eigenfunctions. The new iteration method has superlinear convergence rates and gives a better performance in numerical testing, compared with the other popular methods of rootfinding. Moreover, piecewise particular solutions are used for a basic model of eigenvalue problems on the unit square with the Dirichlet condition. Numerical experiments are also conducted for the eigenvalue problems with singularities. Our new algorithms using piecewise particular solutions are well suited to seek very accurate solutions of eigenvalue problems, in particular those with multiple singularities, interfaces and those on unbounded domains. Using piecewise particular solutions has also the advantage to solve complicated problems because uniform particular solutions may not always exist for the entire solution domain.

1 Introduction
2 New Numerical Algorithms for Eigenvalue Problems
2.1 The Trefftz Method for (1.6)
2.2 Heuristic Ideas of Degeneracy of (1.6)
2.3 New Algorithms for Seeking Eigenvalues and Eigenfunctions
2.4 Solution of Nonlinear Equations
3 Basic Model and Numerical Experiments
3.1 A Basic Sample of Eigenvalue Problems and Particular Solutions
3.2 Description of the Algorithms for TM
3.3 Investigation of Behavior for the Function f(k) as k^2 approach to lamda_{min}
3.4 Computation for the Minimal Eigenvalue and Corresponding Eigenfunction
4 The Eigenvalue Problem for the Cracked Beam
5 Summaries and Discussions

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