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| 研究生： |
鄭月雯 Cheng, Yueh-Wen |
|---|---|
| 論文名稱： |
Computing the Eigenproblem of a Real Orthogonal Matrix |
| 指導教授： |
王太林
Wang, Tai-Lin |
| 學位類別： |
碩士
Master |
| 系所名稱： |
理學院 - 應用數學系 Department of Mathematical Sciences |
| 論文出版年： | 2000 |
| 畢業學年度： | 88 |
| 語文別： | 英文 |
| 論文頁數： | 28 |
| 中文關鍵詞： | 正交矩陣的特徵問題 、Schur參數 、奇異值問題 |
| 外文關鍵詞： | orthogonal eigenproblem, Schur parameters, singular value problem, CLAPACK |
| 相關次數： | 點閱：193 下載：41 |
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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.
封面頁
證明書
致謝詞
論文摘要
目錄
1 Introduction
1.1 Schur parameterization
2 Fundamental principles
2.1 Computation of eigenvalues
2.1.1 Introducing two symmetric tridiagonal matrices C and S
2.1.2 Transforming C and S into bidiagonal form
2.1.3 Singular value computing and numerical stability
2.2 Computation of eigenvectors
2.2.1 Finding the left singular vectors of C
2.2.2 Determining the eigenvectors of C and S
2.2.3 Transforming back to the eigenvectors of H
3 Examples and numerical results
3.1 Tested examples
3.2 Numerical results of eigenvalues
3.3 Numerical results of eigenvectors
3.4 Conclusion
Appendix
References
[1] S. Ammar, W. B. Gragg, L. Reichel, On the Eigenproblem for Orthogonal Matrices, Proc. 25th IEEE Conference on Decision and Control, pp.1963--1966. Athens: Greece (1986).
[2] W. B. Gragg, The QR Algorithm for Unitrary Hessenberg Matrices, J. Comput. Appl. Math. vol. 16, pp.1--8 (1986).
[3] E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. Mckenney, S. Ostrouchov, D. Sorensen, LAPACK Users' Guide, 2nd ed., SIAM, Philadelphia (1995).
[4] J. Demmel, W. Kahan, Computing Small Singular Values of Bidiagonal Matrices With Guaranteed High Relative Accuracy, SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp. 873--912 (1990).
[5] W. B. Gragg, L. Reichel, A Divide and Conquer Method for Unitrary and Orthogonal Eigenproblems, Numer. Math. vol. 57, pp. 695--718 (1990).
[6] G. S. Ammar, L. Reichel, D. C. Sorensen, An Implementation of a Divide and Conquer Algorithm for the Unitrary Eigenproblem, ACM Trans. Math. Softw. vol. 18, no. 3, pp. 292--307 (1992).
[7] T. L. Wang, Lecture Notes on Basic Matrix Eigenproblem Computations with the QR Transformation, unpublished manuscript.
[8] V. F. Pisarenko, The retrieval of harmonics from a covariance function, Geophys. J. R. Astr. Soc. vol. 33, pp. 347--366 (1973).