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Title: Computing the Eigenproblem of a Real Orthogonal Matrix
Authors: 鄭月雯
Cheng, Yueh-Wen
Contributors: 王太林
Wang, Tai-Lin
Cheng, Yueh-Wen
Keywords: 正交矩陣的特徵問題
orthogonal eigenproblem
Schur parameters
singular value problem
Date: 2000
Issue Date: 2016-04-18 16:31:54 (UTC+8)
Abstract: 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.
Reference: [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[7] T. L. Wang, Lecture Notes on Basic Matrix Eigenproblem Computations with the QR Transformation, unpublished manuscript.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).
Description: 碩士
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Data Type: thesis
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