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The QR Algorithm for Eigenvalue Estimation: Theory and Experiments The QR Algorithm for Eigenvalue Estimation: Theory and Experiments Except where reference is made to the work of others, the work described in this thesis is my own or was done in collaboration with my advisory committee. This thesis does not include proprietary or classi¯ed information. Wei Feng Certi¯cate of Approval: Tin-Yau Tam Thomas H. Pate, Chair Professor Professor Mathematics and Statistics Mathematics and Statistics John P. Holmes, III George T. Flowers Professor Dean Mathematics and Statistics Graduate School The QR Algorithm for Eigenvalue Estimation: Theory and Experiments Wei Feng A Thesis Submitted to the Graduate Faculty of Auburn University in Partial Ful¯llment of the Requirements for the Degree of Master of Science Auburn, Alabama December 19, 2008 The QR Algorithm for Eigenvalue Estimation: Theory and Experiments Wei Feng Permission is granted to Auburn University to make copies of this thesis at its discretion, upon the request of individuals or institutions and at their expense. The author reserves all publication rights. Signature of Author Date of Graduation iii Vita Wei Feng, only son and the third child of Zhiguo Feng and Shulan He, was born in Tangshan city, Hebei Province, China on Nov. 28, 1976. He graduated from the First High School of Luanxian in 1996 and was enrolled in Shanghai Jiao Tong University the same year, where he earned both his Bachelor of Engineering and Master of Science degrees in electrical engineering in 2000 and 2003, respectively. He then worked for Magima Inc. in 2003 for several months before he went to the United States with his beloved wife, Shuang Feng. He then entered the PhD program in electrical engineering of Auburn University in Aug. 2004, where he is expected to acquire the degree in Spring, 2009. He was accepted in the Master's program in mathematics of Auburn University in Feb. 2008. He and his wife shared the same high school, the same university for their undergraduate and graduate years in China, and the same university in the United States in Auburn. They are blessed with a lovely daughter, Grace T. Feng, who was born on May 19, 2005. iv Thesis Abstract The QR Algorithm for Eigenvalue Estimation: Theory and Experiments Wei Feng Master of Science, December 19, 2008 (M.E., Shanghai Jiao Tong University, 2003) (B.S., Shanghai Jiao Tong University, 2000) 57 Typed Pages Directed by Thomas H. Pate In this thesis, we explore one of the most important numerical algorithms ever invented, the QR algorithm for computing matrix eigenvalues. First, we describe out notations and mathematical symbols used throughout the thesis in Chapter 1. Then we lay the ground work by stating and proving some basic lemmas in Chapter 2. Then in Chapter 3, we prove the convergence of the QR algorithm under the assumption of distinct magnitudes for all eigenvalues. This constraint is relaxed in Chapter 4, where we prove the convergence of the QR algorithm under the assumption of possibly equal magnitude eigenvalues. Finally, in Chapter 5, we present some numerical experiments to validate the conclusions drawn in this thesis. v Acknowledgments First of all, I would like to thank Dr. Thomas H. Pate, my principle adviser, for his teaching and guidance throughout the whole process. What is not shown in this work is his meticulous yet picture clear methodology toward mathematics that I have witnessed and hopefully learned from. I am also thankful to Dr. Gary M. Sampson, Dr. Jo W. Heath, Dr. Narendra K. Govil, Dr. Amnon J. Meir and Dr. Stephen E. Stuckwisch, from whose teachings I have enjoyed mathematics. I also want to thank Dr. Tin-Yau Tam and Dr. John P. Holms, for sitting on my committee and proof-reading my thesis. I also want to thank Dr. Thomas S. Denney Jr. and Dr. Stanley J. Reeves from the Electrical and Computer Engineering department not only for the ¯nancial support they granted me during the years I spent in Auburn, but also the teaching and guidance that have honed my problem solving skills. I also appreciate the interactions that I had with the students in the image processing group. Finally I want to thank my family members for providing the love and support all the time. I want to thank my wife not only for her love and care, but also for her sharing the burden for the years. I want to thank my baby daughter for being the greatest and happiest gift to me. In the end, I want to thank my parents and my sisters, who have raised me and supported me unconditionally all my life. vi Style manual or journal used Journal of Approximation Theory (together with the style known as \aums"). Bibliography follows van Leunen's A Handbook for Scholars. Computer software used The document preparation package TEX (speci¯cally LATEX) together with the departmental style-¯le aums.sty. vii Table of Contents List of Figures ix 1 Introduction 1 2 Useful Lemmas and Theorems 4 3 Convergence of the QR Algorithm 21 4 Generalization: Equal-magnitude eigenvalues 29 5 Experiments 38 Bibliography 48 viii List of Figures 5.1 Lower triangular (o® diagonal) part of Ak converge to zero . 39 5.2 Diagonal elements of Ak converge to eigenvalues of A . 40 5.3 Upper triangular (o® diagonal) part of Ak converge in magnitude . 40 5.4 Lower triangular (o® diagonal) part of A converge to zero . 41 5.5 Diagonal elements of Ak converge to eigenvalues of A . 42 5.6 Upper triangular (o® diagonal) part of A converge in magnitude . 42 5.7 Lower triangular (o® diagonal) part of A converge to zero . 43 5.8 Schematic illustration of the lower triangular o® diagonal blocks of Ak . 43 5.9 Diagonal elements of Ak do NOT converge to eigenvalues of A . 44 5.10 Convergence of eigenvalue multisets of diagonal blocks of Ak to those of A . 44 5.11 Convergence of QR: A is shifted by (2 + i) in the complex plane . 47 ix Chapter 1 Introduction The eigenvalues of an n £ n matrix A are the roots of its characteristic polynomial det(A ¡ ¸I). The famous Abel-Ru±ni theorem states there is no algebraic formula for the roots of a general polynomial of degree ¯ve or higher. This means that the best method for computing the eigenvalues of a general n £ n (n ¸ 5) matrix is likely to be iterative in nature. The most famous eigenvalue algorithm is the QR algorithm discovered by Francis [3, 4] and Kublanovskaya [5] independently. A convergence analysis of the QR algorithm was given by Wilkinson [6]. A brief sketch of the early days history of the development of the QR algorithm was given by Parlett [13]. In this thesis, we present a complete detailed proof of the convergence of the QR algorithm under mild assumptions. First, the convergence is proved assuming that the magnitudes of all eigenvalues are distinct. Then this assumption is loosened such that the magnitudes of some eigenvalues may be equal, under which the convergence is re-de¯ned and proved. Huang and Tam [14] proved the convergence of the QR algorithm for real matrices with non-real eigenvalues. Since the magnitudes of such matrices are distinct except for the conjugate pairs, they ¯t the loosened assumption. We employ certain standard notations throughout this thesis. We let Cm£n denote the set of m £ n complex matrices. Similarly, Rm£n denotes the set of all m £ n real matrices. Both A(i; j) and Aij refer to the (i; j)th element of matrix A. On the other hand, m£n T A = [aij] signi¯es that the (i; j)th element of A is aij. Given A 2 C , we let A denote the transpose of A and we let A¤ denote the conjugate transpose of A. If A is square and ¡1 invertible, then A denotes the inverse of A. The k £ k identity matrix is denoted as Ik. In circumstances where the dimension is unambiguous, the subscript of a matrix is dropped. For example, if confusion seems unlikely, we may write I instead of Ik. The unit vectors 1 representing the coordinate system of dimension n is denoted as ei, for all i = 1; 2; : : : ; n. T Mathematically, ei = [0;:::; 0; 1; 0;:::; 0] , where the number 1 is the i-th element of ei. The notation diag ([d1; d2; ¢ ¢ ¢ ; dn]) represents the diagonal matrix with d1, d2, :::, dn T on the diagonal in that order. Similarly, if x = [x1; x2; : : : ; xn] (or x = [x1; x2; : : : ; xn]) is a vector, then the diagonal matrix with x on the diagonal is de¯ned as diag (x) = diag ([x1; x2; : : : ; xn]). The notation Dg applied on an n£n matrix A is de¯ned as Dg (A) = diag ([A(1; 1);A(2; 2);:::;A(n; n)]). The zero scalar, vectors and matrices are all repre- sented by 0 for simplicity. The distinction8 should be clear in context. The Kronecker delta > <>1; if i = j function is denoted as ± and ± = . The character ¸ is used exclusively ij ij > :>0; otherwise to represent an eigenvalue. For a scalar a, jaj represents the absolute value of a and a denote the complex conjugate of a. For a matrix A, jAj is the matrix whose elements are the absolute value of the corresponding elements of A. In other words, (jAj)ij = jAijj. The Euclidean norm of a n n vector v in C or R is denoted by kvk2. We reserve the notation jjj¢jjj for matrix norms. For example, jjj¢jjjp is the induced p-norm of a square matrix for p ¸ 1 de¯ned as jjjAjjjp = kAxkp maxx6=0 .
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