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論文名稱 Title |
由兩側正交分解計算數值秩 The two-sided orthogonal decompositions for computing the numerical rank |
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系所名稱 Department |
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畢業學年期 Year, semester |
語文別 Language |
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學位類別 Degree |
頁數 Number of pages |
38 |
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研究生 Author |
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指導教授 Advisor |
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召集委員 Convenor |
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口試委員 Advisory Committee |
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口試日期 Date of Exam |
2014-06-13 |
繳交日期 Date of Submission |
2014-07-04 |
關鍵字 Keywords |
Sylvester矩陣、數值秩、正交分解、子空間 orthogonal decomposition, numerical rank, subspaces, Sylvester matrix |
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統計 Statistics |
本論文已被瀏覽 5750 次,被下載 1150 次 The thesis/dissertation has been browsed 5750 times, has been downloaded 1150 times. |
中文摘要 |
計算矩陣的數值秩廣泛的出現在很多科學計算問題中,奇異值分解是一個求數 值秩的穩定方法。然而對於一些數值秩或數值零核維度很小的矩陣來說,此分解的 計算相對來說要花費很多的計算量與儲存空間。本篇論文我們將研究兩個雙邊正交 分解URV 及ULV,其可代替奇異值分解在求數值秩中扮演的角色。我們也要探 討G.W. Stewart 所提出來的URV 及ULV 演算法,分析每一個步驟以及估計 其計算子空間的計算誤差。最後我們將其演算法應用在求Sylvester 矩陣的的數值 零核維度。 |
Abstract |
Rank-revealing arises in a wide variety of applications in scientific com- puting. The singular value decomposition is considered as the standard rank- revealing method. However, it has heavily computational demand both in time and storage when the rank or the nullity is small. In this thesis we study the URV and ULV decompositions which are alternatives for determining the numerical rank for low rank or low nullity matrices. In particular, we inves- tigate each step in the algorithms, proposed by G.W. Stewart, for computing two-sided orthogonal decompositions, and estimate the error bounds of the computed numerical subspaces of the target matrix. Finally, we implement the algorithm for computing the numerical null space of Sylvester matrices. |
目次 Table of Contents |
Contents Thesis Approval Sheet i Chinese abstract ii English abstract iii List of tables v List of figures vi 1 Introduction 1 2 Preliminary 3 2.1 The uniqueness of QR decomposition . . . . . . . . . . . . . . . . . . 3 2.2 SVD decomposition and fundamental subspaces . . . . . . . . . . . . 6 2.3 The angle between two subspaces . . . . . . . . . . . . . . . . . . . . 8 2.4 Power iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Compute the smallest singular value . . . . . . . . . . . . . . . . . . 11 2.6 Sylvester matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Two-sided orthogonal decompositions 14 3.1 URV decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 ULV decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 The bound of the subspaces . . . . . . . . . . . . . . . . . . . . . . . 18 4 Numerical Results 25 5 References 30 List of Tables 1 Numerical result of low rank with high noise . . . . . . . . . . . . . . 25 2 Numerical result of low rank with middle noise . . . . . . . . . . . . . 25 3 Numerical result of low rank with low noise . . . . . . . . . . . . . . 26 4 Numerical result of high rank with high noise . . . . . . . . . . . . . 26 5 Numerical result of high rank with high noise by our program . . . . 27 6 Numerical result of high rank with middle noise . . . . . . . . . . . . 27 7 Numerical result of high rank with middle noise by our program . . . 27 8 Numerical result of high rank with low noise . . . . . . . . . . . . . . 28 9 Numerical result of high rank with low noise by our program . . . . . 28 10 Numerical result of Sylvester matrix . . . . . . . . . . . . . . . . . . . 28 11 Numerical result of Sylvester matrix from [2] . . . . . . . . . . . . . . 29 List of Figures 1 The projection of two vectors . . . . . . . . . . . . . . . . . . . . . . 3 2 The projection of vector and subspace . . . . . . . . . . . . . . . . . 4 3 The angle of vector and subspace . . . . . . . . . . . . . . . . . . . . 9 4 The angle of two subspaces . . . . . . . . . . . . . . . . . . . . . . . . 10 |
參考文獻 References |
[1] Tony F. Chan (1986), Rank Revealing QR Factorizations, Linear Algebra Appl., 88/89 (1987), pp. 67–82. [2] Robert M. Corless, Stephen M. Watt, and Lihong Zhi (2004), QR Factoring to Compute the GCD of Univariate Approximate Polynomials, IEEE Transactions on Signal Processing, Vol. 52, No. 12. [3] Ricardo D. Fierro (1996), Perturbation analysis for two-sided (or compltet) orthogoal decompositions, SIAM J. Matrix Anal. Appl., Vol. 17, No. 2, pp. 383-400. [4] Ricardo D. Fierro and James R. Bunch (1995), Bounding the subspaces from rank revealing two-sided orthogonal decompositions, SIAM J. Matrix Anal. Appl., Vol. 16, No. 3, pp. 743-759. [5] Ricardo D. Fierro and Per Christian Hansen (1997), Low-rank revealing UTV decompositions, Numerical Algorithms, 15, pp. 37–55. [6] Dan Kalman (2002), A Singularly Valuable Decomposition: The SVD of a Matrix, The American University, Washington, DC 20016. [7] Tsung-Lin Lee, Tien-Yien Li, and Zhonggang Zeng (2009), A rank-revealing method with updating, downdatin, and applications. part II, SIAM J. Matrix Anal. Appl., Vol. 31, No. 2, pp. 503-525. [8] F. Lorenzelli, P. C. Hansen, T. F. Chan, and K. Yao (1994), A Systolic Implementation of the Chan/Foster RRQR Algorithm, IEEE Transactions on Signal Processing, Vol. 42, No. 8. [9] Haesun Park and Lars Elden (1995), Downdating the rank-revealing URV decomposition, SIAM J. Matrix Anal. Appl., Vol. 16, No. 1, pp. 138-155. [10] G. W. Stewart (1991), Updating a rank-revealing ULV decomposition, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 494–499. [11] Z. Zeng (2011), The Numerical Greatest Common Divisor of Univariate Polynomials, Contemp. Math., 556, pp. 187-217. |
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