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題名 Computing the Eigenproblem of a Real Orthogonal Matrix 作者 鄭月雯
Cheng, Yueh-Wen貢獻者 王太林
Wang, Tai-Lin
鄭月雯
Cheng, Yueh-Wen關鍵詞 正交矩陣的特徵問題
Schur參數
奇異值問題
orthogonal eigenproblem
Schur parameters
singular value problem
CLAPACK日期 2000 上傳時間 18-Apr-2016 16:31:54 (UTC+8) 摘要 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.參考文獻 [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 [7] T. L. Wang, Lecture Notes on Basic Matrix Eigenproblem Computations with the QR Transformation, unpublished manuscript.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). 描述 碩士
國立政治大學
應用數學系
87751005資料來源 http://thesis.lib.nccu.edu.tw/record/#A2002001743 資料類型 thesis dc.contributor.advisor 王太林 zh_TW dc.contributor.advisor Wang, Tai-Lin en_US dc.contributor.author (Authors) 鄭月雯 zh_TW dc.contributor.author (Authors) Cheng, Yueh-Wen en_US dc.creator (作者) 鄭月雯 zh_TW dc.creator (作者) Cheng, Yueh-Wen en_US dc.date (日期) 2000 en_US dc.date.accessioned 18-Apr-2016 16:31:54 (UTC+8) - dc.date.available 18-Apr-2016 16:31:54 (UTC+8) - dc.date.issued (上傳時間) 18-Apr-2016 16:31:54 (UTC+8) - dc.identifier (Other Identifiers) A2002001743 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/85501 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用數學系 zh_TW dc.description (描述) 87751005 zh_TW dc.description.abstract (摘要) 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。 zh_TW dc.description.abstract (摘要) 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. en_US dc.description.tableofcontents 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 zh_TW dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#A2002001743 en_US dc.subject (關鍵詞) 正交矩陣的特徵問題 zh_TW dc.subject (關鍵詞) Schur參數 zh_TW dc.subject (關鍵詞) 奇異值問題 zh_TW dc.subject (關鍵詞) orthogonal eigenproblem en_US dc.subject (關鍵詞) Schur parameters en_US dc.subject (關鍵詞) singular value problem en_US dc.subject (關鍵詞) CLAPACK en_US dc.title (題名) Computing the Eigenproblem of a Real Orthogonal Matrix en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) [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). zh_TW
