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.

設H是一個實數正交的矩陣，我們要求它的特徵值以及特徵向量。H可以表示成Schur參數的形式。根據Ammar，Gragg及Reichel的論文，我們把H的特徵問題轉換成兩個元素由Schur參數決定的二對角矩陣的奇異值(奇異向量)的問題。我們用這個方法寫成程式並且與CLAPACK的程式比較準確度及速度。最後列出一些數值的結果作為結論。 / Let H be an orthogonal Hessenberg matrix whose eigenvalues, and possibly eigenvectors, are to be determined. Then H can be represented in Schur parametric form [2]. Following Ammar, Gragg, and Reichel's paper [1], we compute the eigenproblem of H by finding the singular values (and vectors) of two bidiagonal matrices whose elements are explicitly known functions of the Schur parameters. We compare the accuracy and speed of our programs using the method described aboved with those in CLAPACK. Numerical results conclude this thesis.