Panju
The Waterloo Mathematics Review
9
Iterative Methods for Computing Eigenvalues and Eigenvectors
Maysum Panju University of Waterloo mhpanju@math.uwaterloo.ca
Abstract: We examine some numerical iterative methods for computing the eigenvalues and eigenvectors of real matrices. The five methods examined here range from the simple power iteration method to the more complicated QR iteration method. The derivations, procedure, and advantages of each method are briefly discussed.
1
Introduction
Eigenvalues and eigenvectors play an important part in the applications of linear algebra. The naive method of finding the eigenvalues of a matrix involves finding the roots of the characteristic polynomial of the matrix. In industrial sized matrices, however, this method is not feasible, and the eigenvalues must be obtained by other means. Fortunately, there exist several other techniques for finding eigenvalues and eigenvectors of a matrix, some of which fall under the realm of iterative methods. These methods work by repeatedly refining approximations to the eigenvectors or eigenvalues, and can be terminated whenever the approximations reach a suitable degree of accuracy. Iterative methods form the basis of much of modern day eigenvalue computation. In this paper, we outline five such iterative methods, and summarize their derivations, procedures, and advantages. The methods to be examined are the power iteration method, the shifted inverse iteration method, the Rayleigh quotient method, the simultaneous iteration method, and the QR method. This paper is meant to be a survey of existing algorithms for the eigenvalue computation problem. Section 2 of this paper provides a brief review of some of the linear algebra background required to understand the concepts that are discussed. In Section 3, the iterative methods are each presented, in order of complexity, and are studied in brief detail. Finally, in Section 4, we provide some concluding remarks and mention some of
9
Iterative Methods for Computing Eigenvalues and Eigenvectors
Maysum Panju University of Waterloo mhpanju@math.uwaterloo.ca
Abstract: We examine some numerical iterative methods for computing the eigenvalues and eigenvectors of real matrices. The five methods examined here range from the simple power iteration method to the more complicated QR iteration method. The derivations, procedure, and advantages of each method are briefly discussed.
1
Introduction
Eigenvalues and eigenvectors play an important part in the applications of linear algebra. The naive method of finding the eigenvalues of a matrix involves finding the roots of the characteristic polynomial of the matrix. In industrial sized matrices, however, this method is not feasible, and the eigenvalues must be obtained by other means. Fortunately, there exist several other techniques for finding eigenvalues and eigenvectors of a matrix, some of which fall under the realm of iterative methods. These methods work by repeatedly refining approximations to the eigenvectors or eigenvalues, and can be terminated whenever the approximations reach a suitable degree of accuracy. Iterative methods form the basis of much of modern day eigenvalue computation. In this paper, we outline five such iterative methods, and summarize their derivations, procedures, and advantages. The methods to be examined are the power iteration method, the shifted inverse iteration method, the Rayleigh quotient method, the simultaneous iteration method, and the QR method. This paper is meant to be a survey of existing algorithms for the eigenvalue computation problem. Section 2 of this paper provides a brief review of some of the linear algebra background required to understand the concepts that are discussed. In Section 3, the iterative methods are each presented, in order of complexity, and are studied in brief detail. Finally, in Section 4, we provide some concluding remarks and mention some of
References: [GL87] Alan George and Joseph W.H. Liu, Householder reflections versus givens rotations in sparse orthogonal decomposition, Linear Algebra and its Applications 88–89 (1987), 223–238. [Ost59] A. M. Ostrowski, On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. iii, Archive for Rational Mechanics and Analysis 3 (1959), 325–340, 10.1007/BF00284184. [QRA] Lecture 28: QR Algorithm, University of British Columbia CS 402 Lecture Notes, 1–13. [Rut69] Heinz Rutishauser, Computational aspects of F. L. Bauer’s simultaneous iteration method, Numerische Mathematik 13 (1969), 4–13, 10.1007/BF02165269.