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研究生： 李仁安
Li, Jen-An
論文名稱： 根據QR分解計算特徵值
Computing Eigenvalues Based on QR factorization
指導教授： 陳朝欽
Chen, Chaur-Chin
口試委員: 黃仲陵
Huang, Chung-Lin
張隆紋
Chang, Long-Wen
學位類別： 碩士
Master
系所名稱： 電機資訊學院 - 資訊工程學系
Computer Science
論文出版年： 2021
畢業學年度： 109
語文別： 英文
論文頁數： 20
中文關鍵詞： 特徵值 、QR分解 、QR演算法
外文關鍵詞： QR factorization, Householder transformation, Givens rotation, Hessenberg matrix, QR algorithm
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    • QR 分解是數值線性代數中使用最廣泛的矩陣分解方法之一。它將矩陣 A 分解為A = QR，其中 Q 是正交矩陣，R 是上三角矩陣。 QR 分解有很多應用，例如用來計算矩陣的特徵值。
    在本論文中，我們首先回顧了三種 QR 分解方法。 接下來，我們實作如何用QR分解計算特徵值。 我們透過將原始矩陣轉換為 Hessenberg 矩陣來降低之後每一次QR分解所花費的時間。再來，我們通過對 Hessenberg 矩陣使用Rayleigh shift來加快收斂速度。 但是我們發現有些矩陣最後不會收斂到上三角矩陣。 所以我們應用另一個Wilkinson shift來解決這個問題。

    QR factorization, sometimes also called QR decomposition, is one of the most widely used matrix decomposition method in numerical linear algebra. It decomposed a matrix A into a product A = QR, where Q is an orthogonal matrix and R is an upper triangular matrix. QR factorization has many applications such as QR algorithm, which is used to compute the eigenvalues of a matrix.
    In this thesis, we first review three QR factorization methods. Next, we implement the QR algorithm. We reduce the cost of QR factorization per iteration by transforming an original matrix to a Hessenberg matrix. And we speed up the convergence rate by applying a Rayleigh quotient shift to a Hessenberg matrix. But we find some matrix will not converge to an upper triangular matrix at the end. So we apply another shift called Wilkinson shift to overcome this problem.

    Chapter 1 Introduction …………………………………………………………………………………………… 1 Chapter 2 Background Review for QR Factorization ……………………… 3 2.1 Gram-Schmidt ………………………………………………………………………………………………………………… 3 2.2 Givens Rotation ……………………………………………………………………………………………………… 4 2.3 Householder Transformation ………………………………………………………………………… 8 Chapter 3 Computing Eigenvalues Based on QR Factorization …………… 11 3.1 Basic Algorithm ………………………………………………………………………………………………………… 11 3.2 A Convertion to Hessenberg Matrix …………………………………………………………… 12 3.3 Hessenberg QR Algorithm with Shift ……………………………………………………… 13 3.4 Different Shift Strategies …………………………………………………………………………… 14 Chapter 4 Experimental Result ………………………………………………………………………… 16 4.1 A 4 by 4 Matrix Example …………………………………………………………………………………… 16 4.2 Average Iterations of Different Strategies …………………………………… 17 Chapter 5 Conclusion ………………………………………………………………………………………………… 18 References …………………………………………………………………………………………………………………………………………… 19

    References
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    [Fran1961] J.G.F. Francis, The QR transformation, a unitary analogue to the LR transformation-part 1, Computer Journal, 265-274, 1961.
    [Jian1985] E. Jiang and Z. Zhang, A new shift of the QL algorithm for irreducible symmetric tridiagonal matrices, Linear Algebra Applications, 261-272, 1985.
    [Kubl1961] V.N. Kublanovskaya, Certain algorithms for the solution of the complete eigenvalue problem, Doklady Akademii Nauk, 26-28, 1961.
    [Parl1998] B.N. Parlett, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, PA, 151-187, 1998.
    [Ruti1958] H. Rutishauser, Solution of eigenvalue problems with the LR-transformation, National Bureau of Standards Applied Math Series, 47-80, 1958.
    [Stra2006] G. Strang, Linear Algebra and Its Applications, Brooks/Cole, Cengage Learning 4e, 2006.
    [Wilk1968] J.H. Wilkinson, Global convergence of tridiagonal QR algorithm with origin shifts, Linear Algebra Applications, 409-420, 1968.
    [Web01] Abel-Ruffini theorem, https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem, last access on June 30, 2021.
    [Web02] Schur-decomposition, https://en.wikipedia.org/wiki/Schur_decomposition, last access on June 30, 2021.
    [Web03] B.A. Cipra, The best of the 20th Century: Editors Name Top 10 Algorithms, https://archive.siam.org/pdf/news/637.pdf, last access on June 30, 2021.

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