本文主旨在探討計算正交矩陣特徵值問題的三種方法。首先利用正交矩陣之特質，將 該矩陣以蘇爾參數(Schur parameters)之形式表出。再以下列三種計算特徵值問題的 方法，去比較這些方法所算出之特徵至與特徵值與特徵向量之精確度。 1.QR方法。 2.DC方法1。 3.SC方法2。 最後，並以此三種方法之適用性作為結論。 / ABSTRACT The main topic of this essay is to discuss three methods which compute the eigen-problem of orthogonal matrices. First, we use the characteristic of orthogonal matrix to represent the matrix with Schur parmetric form. Then we use the following methods based on structure above to compute and compare the accuracy of eigenvalues andeigenvectors.

用QR與LR迭代法求矩陣特徵值與特徵向量之過程中，前人曾提出位移策略以加速其收斂速度，其中最有效的是Wilkinson 移位值。在此我們希望尋求能使收斂更快速的位移值。 我們首先嘗使用一三階子矩陣之特徵值作為一次QR迭代之移位值。在此子矩陣之特徵值中，我們選擇最接近Wilkinson 移位值的特徵值為移位值，期使特徵值之收斂更快。 另一移位策略是用一較快速省功的算則先計算矩陣之特徵值，再以這些計算值作為QR迭代之位移值，來計算較為費功的特徵向量。希望能較快得到所需要的特徵值與特徵向量。 在計算特徵值之算則中，Cholesky迭代法以其計算簡單，執行速度快為我們所選擇。由程式執行結果可知這兩種算則較EISPACK 的算則分別節省了約10% 與30% 的運算量。我們比較這些策略，並將結果列於文中。 / Abstract The QR and LR algorithms are the general methods for computing eigenvalues and eigenvectors of a dense matrix. In this paper, we propose some shift strategies that can increase the efficiency of the QR algorithm by first computing the eigenvalues of the matrix (or its trailing submatrix) in a fast and economical way, and then using them as shifts to find the eigenvalues and their corresponding eigenvectors. When incorporated with QR algorithm, these kinds of shift strategies can save about 10 to 30percent of work in arithmetic operations.

領域詞典包含許多專業的詞彙以及對詞彙的定義，但詞典中詞彙間的關係是被隱藏起來的，本研究運用自然語言處理的相關技術，提出運用領域詞典找出詞彙間關係建構特定領域語義網路的方法。 / A domain dictionary contains many professional words and their definitions. In general, there are many hidden relations among words in a dictionary. In this thesis, we use techniques of natural language processing to find out these relations, and bring up a method to construct a domain specific lexical semantic network.

給定兩個隨機變數的條件機率矩陣A和B，相容性問題的主要課題包 含：(一)如何判斷他們是否相容？若相容，則如何檢驗聯合分配的唯一性 或找出所有的聯合分配；(二)若不相容，則如何訂定測量不相容程度的方 法並找出最近似聯合分配。目前的文獻資料有幾種解決問題的途徑，例 如Arnold and Press (1989)的比值矩陣法、Song et al. (2010)的不可約 化對角塊狀矩陣法及Arnold et al. (2002)的數學規劃法等，經由這些方法 的啟發，本文發展出創新的特徵向量法來處理前述的相容性課題。 當A和B相容時，我們觀察到邊際分配分別是AB′和B′A對應特徵值1的 特徵向量。因此，在以邊際分配檢驗相容性時，特徵向量法僅需檢驗滿足 特徵向量條件的邊際分配，大幅度減少了檢驗的工作量。利用線性代數中 的Perron定理和不可約化對角塊狀矩陣的概念，特徵向量法可圓滿處理相 容性問題(一)的部份。 當A和B不相容時，特徵向量法也可衍生出一個測量不相容程度的簡單 方法。由於不同的測量方法可得到不同的最近似聯合分配，為了比較其優 劣，本文中提出了以條件分配的偏差加上邊際分配的偏差作為評量最近似 聯合分配的標準。特徵向量法除了可推導出最近似聯合分配的公式解外， 經過例子的驗證，在此評量標準下特徵向量法也獲得比其他測量法更佳的 最近似聯合分配。由是，特徵向量法也可用在處理相容性問題(二)的部份。 最後，將特徵向量法實際應用在兩人零和有限賽局問題上。作業研究的 解法是將雙方採取何種策略視為獨立，但是我們認為雙方可利用償付值表 所提供的資訊作為決策的依據，並將雙方的策略寫成兩個條件機率矩陣， 則賽局問題被轉換為相容性問題。我們可用廣義相容的概念對賽局的解進 行分析，並在各種測度下討論賽局的解及雙方的最佳策略。 / Given two conditional probability matrices A and B of two random variables, the issues of the compatibility include: (a) how to determine whether they are compatible? If compatible, how to check the uniqueness of the joint distribution or find all possible joint distributions; (b) if incompatible, how to measure how far they are from compatibility and find the most nearly compatible joint distribution. There are several approaches to solve these problems, such as the ratio matrix method(Arnold and Press, 1989), the IBD matrix method(Song et al., 2010) and the mathematical programming method(Arnold et al., 2002). Inspired by these methods, the thesis develops the eigenvector approach to deal with the compatibility issues. When A and B are compatible, it is observed that the marginal distributions are eigenvectors of AB′ and B′A corresponding to 1, respectively. While checking compatibility by the marginal distributions, the eigenvector approach only checks the marginal distributions which are eigenvectors of AB′ and B′A. It significantly reduces the workload. By using Perron theorem and the concept of the IBD matrix, the part (a) of compatibility issues can be dealt with the eigenvector approach. When A and B are incompatible, a simple way to measure the degree of incompatibility can be derived from the eigenvector approach. In order to compare the most nearly compatible joint distributions given by different measures, the thesis proposes the deviation of the conditional distributions plus the deviation of the marginal distributions as the most nearly compatible joint distribution assessment standard. The eigenvector approach not only derives formula for the most nearly compatible distribution, but also provides better joint distribution than those given by the other measures through the validations under this standard. The part (b) of compatibility issues can also be dealt with the eigenvector approach. Finally, the eigenvector approach is used in solving game problems. In operations research, strategies adopted by both players are assumed to be independent. However, this independent assumption may not be appropriate, since both players can make decisions through the information provided by the payoffs for the game. Let strategies of both players form two conditional probability matrices, then the game problems can be converted into compatibility issues. We can use the concept of generalized compatibility to analyze game solutions and discuss the best strategies for both players in a variety of measurements.