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| Title: | 移位QR算則在三對角矩陣上之收斂
Convergence of the Shifted QR Algorithm on Tridiagonal Matrices |
| Authors: | 蔡淑芬
Tsai ,Shu-Fen |
| Contributors: | 王太林
Wang ,Tai-Lin
蔡淑芬
Tsai ,Shu-Fen |
| Keywords: | QR Algorithm
Tridiagonal |
| Date: | 2003 |
| Issue Date: | 2009-09-18 18:28:39 (UTC+8) |
| Abstract: | 在計算矩陣的特徵值(eigenvalues)中,QR演算法是一種常見的技巧. 尤其如果使用適當的移位,將可以較快得到特徵值. 在本文中提出一種新的移位策略, 我們證明這各方法是可行的,而且可適用於任何矩陣. 換句說, 本篇論文主旨即是提出有關新的移位QR演算法的收斂.
The QR algorithm is a popular method for computing all the
eigenvalues of a dense matrix. If we use a proper shift, we can
accelerate convergence of the iterative process. Hence, we design a new shift strategy which includes an eigenvalue of the trailing principal 3-by-3 submatrix of the tridiagonal matrix. We prove the global convergence of the new strategy. In other words, the purpose of this thesis is to propose a theory of the convergence of a new shifted QR algorithm.
Abstract i
中文摘要 ii
1 Introduction 1
2 Preliminaries 2
2.1 Notation 2
2.2 The shifted QR algorithm 2
2.3 Shift strategies 4
2.4 The convergence of sequences 5
3 A Residual Estimate 5
4 Convergence of the QR Iteration 8
5 Conclusions and Future Work 11
Reference 12
Appendix 14 |
| Reference: | T. K. Dekker and J. F. Traub, The shifted QR algorithm for Hermitian matrices, Linear Algebra Appl., 4 (1971), pp. 137-154.
James Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997.
K. Gates and W. B. Gragg, Notes on TQR algorithms, J. Comput. Appl. Math., 86 (1997), pp. 195-203.
G. H. Golub and C. F. Van Loan, Matrix Computations}, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
W. Hoffmann, B.N. Parlett, A new proof of global convergence for
the tridiagonal QL algorithm}, SIAM J. Numer. Anal., 15 (1978), pp. 929-937.
E. Jiang and Z. Zhang, A new shift of the QL algorithm for irreducible symmetric tridiagonal matrices}, Linear Algebra Appl., 65 (1985), pp. 261-272.
B. N. Parlett, The Symmetric Eigenvalue Problem, revised ed., SIAM, Philadelphia, PA, 1998.
Y. Saad, Shifts of origin for the QR algorithm, Proceedings IFIP
Congress, Toronto, 1974.
G. Thomas and R. Finney, Calculus and Analytic Geometry, 9th ed., Addison-Wesley publishing company, 1996.
T.-L. Wang, Convergence of the tridiagonal QR algorithm, Linear Algebra Appl., 322 (2001), pp. 1-17.
J. H. Wilkinson, Global convergence of tridiagonal QR algorithm
with origin shifts, Linear Algebra Appl., 1 (1968), pp. 409-420. |
| Description: | 碩士
國立政治大學
應用數學研究所
90751008
92 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0090751008 |
| Data Type: | thesis |
| Appears in Collections: | [應用數學系] 學位論文
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