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論文名稱 Title |
矩陣QR分解揭示低秩與高秩演算法的探討 A study on low-rank and high-rank revealing QR algorithms |
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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 |
43 |
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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 |
2018-06-28 |
繳交日期 Date of Submission |
2018-08-14 |
關鍵字 Keywords |
QR 分解、奇異值、數值秩 QR factorization, singular value, numerical rank |
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統計 Statistics |
本論文已被瀏覽 5693 次,被下載 34 次 The thesis/dissertation has been browsed 5693 times, has been downloaded 34 times. |
中文摘要 |
低秩或高秩矩陣的秩揭示問題在科學計算中廣泛存在,例如找低秩近似矩陣、計算主成分分析的主要子空間、非滿秩最小平方和問題或者檢測同倫法中的曲線結構等等,雖然可以應用 Golub-Reinsch SVD 算法計算所有的奇異值,從而揭示秩,但在這些情況下相對昂貴。由 Tony Chan 和 P. C. Hansen 提出的 RRQR 和 L-RRQR 算法,當 rank gap 足夠大且矩陣是高秩或低秩時,在揭示秩方面是相對有效和可靠的。而這兩個演算法的準確性和對應於最大或最小奇異值的奇異向量估計有很大的相關。在這篇論文中我們研究奇異向量的估計並更正了一個證明。 |
Abstract |
The rank revealing problem for low rank or low nullity matrices arises widely in scientific computing, such as finding a low rank approximation to a matrix, computing the dominate subspace in principal component analysis, solving rank deficient least squares problems, and detecting the curve structure in homotopy continuation method. Although the Golub-Reinsch SVD algorithm can be applied to calculate all singular values and thus reveal the rank, it becomes relatively expensive in these situations. RRQR and L-RRQR algorithm, proposed by Tony Chan and P. C. Hansen, are efficient and reliable methods for rank revealing when the rank gap is well-defined and the matrix is of low rank or low nullity. The accuracy of the methods heavily depends on the estimation of singular vector corresponding to the largest or smallest singular value. In this thesis we study the singular vector estimation and correct a proof in a deflation theorem. |
目次 Table of Contents |
[論文審定書, i] [摘要, ii] [Abstract, iii] [1 Introduction ,1] [2 Preliminary, 5] [3 Rank Revealing QR Factorization for High Rank Matrix, 8] [3.1 High-rank Revealing QR algorithm, 8] [3.2 Compute the smallest singular vector ,10] [3.3 Deflation process in RRQR , 11] [4 Rank Revealing QR Factorization for Low Rank Matrix , 13] [4.1 Low-rank Revealing QR algorithm , 13] [4.2 Compute the biggest singular value , 15] [4.3 Deflation process in L-RRQR , 17] [5 Numerical Experiments, 18] [5.1 Behavior of power iteration and inverse iteration, 18] [5.2 The distribution of diagonal elements in QR, 19] [5.3 Comparison of QR, OCP, and RRQR ,24] [5.4 Comparison of QR, OCP, and L-RRQR , 26] [5.5 Conclusion , 28] [6 Applications , 29] [6.1 Finding low rank approximation, 29] [6.2 Solving the rank deficient least squares problem, 29] [6.3 Subset selection problems , 30] [7 Appendix 32] [7.1 Pseudoinverse of a triangular matrix , 32] [7.2 A posteriori bounds on singular values , 34] |
參考文獻 References |
[1] C. H. Bischof and G. Quintana-Orti, Algorithm 782: Codes for rank-revealing QR factorizations of dense matrices, ACM Trans. Math. Software, 24, pp. 254–257, 1998. [2] Bj¨orck, Numerical Methods for Least Squares Problems, SIAM, Philadelphia, 1996. [3] Tony F. Chan, Rank Revealing QR Factorizations, Lin. Alg. Appl., 88/89, pp. 67–82, 1987. [4] T. R. Chan and P. C. Hansen, Low-rank revealing QR factorizations, Numer. Lin. Alg. Appl., 1, pp. 33–44, 1994. [5] B. H. Dayton and Z. Zeng, Computing the multiplicity structure in solving polynomial systems, Proceedings of ISSAC ‘05, ACM Press, pp. 116–123, 2005. [6] Z. Drmaˇc and Z. Bujanovi´c, On the Failure of Rank-Revealing QR Factorization Software - A Case Study, ACM Trans. Math. Software, 35, Article 12, 2008. [7] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins Univ. Press, Baltimore, MD, 1996. [8] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985. [9] R. Hanson and C. Lawson, Solving Least Squares Problems, Prentice-Hall, 1974. [10] T. M. Hwang, W. W. Lin and E. K. Yang, Rank-revealing LU Factorization, Lin. Alg. Appl., 175, pp. 115–141, 1992. [11] C. T. Pan, On the existence and computation of rank-revealing LU factorizations, Lin. Alg. Appl., 316, pp. 199–222, 2000. 31 |
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