## Computing An Eigenvector With Inverse Iteration (1997)

Venue: | SIAM REVIEW |

Citations: | 41 - 1 self |

### BibTeX

@ARTICLE{Ipsen97computingan,

author = {Ilse C. F. Ipsen},

title = {Computing An Eigenvector With Inverse Iteration},

journal = {SIAM REVIEW},

year = {1997},

volume = {39},

pages = {254--291}

}

### Years of Citing Articles

### OpenURL

### Abstract

The purpose of this paper is two-fold: to analyse the behaviour of inverse iteration for computing a single eigenvector of a complex, square matrix; and to review Jim Wilkinson's contributions to the development of the method. In the process we derive several new results regarding the convergence of inverse iteration in exact arithmetic. In the case of normal matrices we show that residual norms decrease strictly monotonically. For eighty percent of the starting vectors a single iteration is enough. In the case of non-normal matrices, we show that the iterates converge asymptotically to an invariant subspace. However the residual norms may not converge. The growth in residual norms from one iteration to the next can exceed the departure of the matrix from normality. We present an example where the residual growth is exponential in the departure of the matrix from normality. We also explain the often significant regress of the residuals after the first iteration: it occurs when the no...

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Citation Context ...ion. It is used as a criterion for terminating the iterations. Once the residual is small enough, inverse iteration stops because then and x k are an eigenpair of a nearby matrix: Theorem 2.1 (x15 in =-=[18]-=-). Let A be a complex, square matrix, and let r k = (A \Gamma I)x k be the residual for some numbersand vector x k with kx k k = 1. Then there is a matrix E k with (A + E k \Gamma I)x k = 0 and kE k k... |

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Citation Context ..., x1.2]. Although N is not unique, kNkF is. Here we are interested in the two-norm, kNk, which is not unique. But since Frobenius and two norms are related by [17, x2.2] 1 p n kNkFskNkskNkF ; 21 e.g. =-=[15, 22, 35, 36]-=- 21 we are content to know that kNk is at most p n away from a unique bound. We measure the departure of A \Gamma I from normality by j j kNk k( \GammasI) \Gamma1 k = kNk=ffl: This is a relative measu... |

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Citation Context ...radual development of ideas over several papers makes it difficult to realise what Wilkinson 2 [11, x5.4], [41, xx4.6-9], [40], [46, xIV.1.3] [62, x3] 3 [10, x3], [11, x5.9], [38, x2, x3], [45, x4] 4 =-=[10, 29, 43, 8]-=- 5 [44, 45, 57, 62, 63] 3 has accomplished. Therefore we decided to compile and order his main results and to set out his ideas. Wilkinson's numerical intuition provided him with many insights and emp... |

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Citation Context ... C A 33 be of order m ands6= 0. Then kJ \Gamma1 ks(1 + jj) m\Gamma1 jj m : Now we bound the norm of a matrix consisting of several Jordan blocks. Theorem 8.2 (Proposition 1.12.4 in [11], Theorem 8 in =-=[31]-=-). Let J be a Jordan matrix whose Jordan blocks have diagonal elementssi ; let ffl j min i j i j; and let m be the order of a Jordan block J j for which kJ \Gamma1 k = kJ \Gamma1 j k. If ffl ? 0 then ... |

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Citation Context ... of ideas over several papers makes it difficult to realise what Wilkinson 2 [11, x5.4], [41, xx4.6-9], [40], [46, xIV.1.3] [62, x3] 3 [10, x3], [11, x5.9], [38, x2, x3], [45, x4] 4 [10, 29, 43, 8] 5 =-=[44, 45, 57, 62, 63]-=- 3 has accomplished. Therefore we decided to compile and order his main results and to set out his ideas. Wilkinson's numerical intuition provided him with many insights and empirical observations for... |

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Citation Context ...nitude in the original matrix A \Gamma I [58, x2]. For Gaussian elimination with partial pivoting aes2 n\Gamma1 [59, xx8, 29]. Although one can find practical examples with exponential growth factors =-=[16]-=-, numerical experiments with random matrices suggest growth factors proportional to n 2=3 [53]. According to public opinion, ae is small [17, x3.4.6]. Regarding the possibility of large elements in U ... |

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Citation Context ... 1.1],[7, Definition 9.1], HeF (A) j kA A \Gamma AA kF kA 2 kFs2 kAk 2 F kA 2 kF ; is a lower bound for the two-norm eigenvector condition number of a diagonalisable matrix: Theorem 9.3 (Theorem 8 in =-=[49]-=- adapted to the two-norm). Let A be n \Theta n. If A is diagonalisable with eigenvector matrix V then (V ) 4s1 + 1 2 HeF (A) 2 : Acknowledgements. I thank Carl Meyer for encouraging me to write this p... |

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Citation Context ..., x1.2]. Although N is not unique, kNkF is. Here we are interested in the two-norm, kNk, which is not unique. But since Frobenius and two norms are related by [17, x2.2] 1 p n kNkFskNkskNkF ; 21 e.g. =-=[15, 22, 35, 36]-=- 21 we are content to know that kNk is at most p n away from a unique bound. We measure the departure of A \Gamma I from normality by j j kNk k( \GammasI) \Gamma1 k = kNk=ffl: This is a relative measu... |

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Citation Context ...radual development of ideas over several papers makes it difficult to realise what Wilkinson 2 [11, x5.4], [41, xx4.6-9], [40], [46, xIV.1.3] [62, x3] 3 [10, x3], [11, x5.9], [38, x2, x3], [45, x4] 4 =-=[10, 29, 43, 8]-=- 5 [44, 45, 57, 62, 63] 3 has accomplished. Therefore we decided to compile and order his main results and to set out his ideas. Wilkinson's numerical intuition provided him with many insights and emp... |

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Citation Context ...for computing eigenvectors corresponding to selected eigenvalues which have already been computed more or less accurately.' A look at software in the public domain shows that this is still true today =-=[1, 44, 47]-=-. The purpose of this paper is two-fold: to analyse the behaviour of inverse iteration; and to review Jim Wilkinson's contributions to the development of inverse iteration. Although inverse iteration ... |

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Citation Context ...e iteration, departure from normality, illconditioned linear system AMS subject classification. 15A06, 15A18, 15A42, 65F15 1. Introduction. Inverse Iteration was introduced by Helmut Wielandt in 1944 =-=[56]-=- as a method for computing eigenfunctions of linear operators. Jim Wilkinson turned it into a viable numerical method for computing eigenvectors of matrices. At present it is the method of choice for ... |

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Citation Context ...radual development of ideas over several papers makes it difficult to realise what Wilkinson 2 [11, x5.4], [41, xx4.6-9], [40], [46, xIV.1.3] [62, x3] 3 [10, x3], [11, x5.9], [38, x2, x3], [45, x4] 4 =-=[10, 29, 43, 8]-=- 5 [44, 45, 57, 62, 63] 3 has accomplished. Therefore we decided to compile and order his main results and to set out his ideas. Wilkinson's numerical intuition provided him with many insights and emp... |

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Citation Context ...ented it in 1958 [57] as a result of trying to improve Givens's method for computing a single eigenvector of a symmetric tridiagonal matrix (a different improvement of Givens's method is discussed in =-=[42]-=-). We modify Wilkinson's idea slightly and present it for a general complex matrix A rather than for a real symmetric tridiagonal matrix. His idea is the following: Ifsis an eigenvalue of the n \Theta... |

2 |
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Citation Context ...l eigenvalues are available. It is frequently used in structural mechanics, for instance, to determine extreme eigenvalues and corresponding eigenvectors of Hermitian positive-(semi)definite matrices =-=[2, 3, 20, 21, 28, 48]-=-. Suppose we are given a real or complex square matrix A and an approximationsto an eigenvalue of A. Inverse iteration generates a sequence of vectors x k from a given starting vector x 0 by solving t... |

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Citation Context ... residual decreases with ill-conditioning of the eigenvectors. Hence the residual can be much smaller than the accuracy of the shift if the eigenvectors are ill-conditioned. Theorem 4.1 (Theorem 1 in =-=[5]-=-). Let A be a diagonalisable matrix with eigenvector matrix V , and let r k = (A \GammasI)x k be the residual for some numbersand vector x k with kx k k = 1. Then kr k ksffl=(V ): Proof. Ifsis an eige... |

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Citation Context ...dix 1: Facts about Jordan Blocks. In this section we give bounds on the norm of the inverse of a Jordan block. First we give an upper bound for a single Jordan block. Lemma 8.1 (Proposition 1.12.4 in =-=[11]-=-). Let J = 0 B B B @s1 . . . . . . 1 1 C C C A 33 be of order m ands6= 0. Then kJ \Gamma1 ks(1 + jj) m\Gamma1 jj m : Now we bound the norm of a matrix consisting of several Jordan blocks. Theorem 8.2 ... |

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Citation Context ...l eigenvalues are available. It is frequently used in structural mechanics, for instance, to determine extreme eigenvalues and corresponding eigenvectors of Hermitian positive-(semi)definite matrices =-=[2, 3, 20, 21, 28, 48]-=-. Suppose we are given a real or complex square matrix A and an approximationsto an eigenvalue of A. Inverse iteration generates a sequence of vectors x k from a given starting vector x 0 by solving t... |

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Citation Context ...l eigenvalues are available. It is frequently used in structural mechanics, for instance, to determine extreme eigenvalues and corresponding eigenvectors of Hermitian positive-(semi)definite matrices =-=[2, 3, 20, 21, 28, 48]-=-. Suppose we are given a real or complex square matrix A and an approximationsto an eigenvalue of A. Inverse iteration generates a sequence of vectors x k from a given starting vector x 0 by solving t... |

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Citation Context ...n process can be expensive. A detailed discussion of these issues can be found for instance in [44, 8]. 1.1. Motivation. This paper grew out of commentaries about Wielandt's work on inverse iteration =-=[26]-=- and the subsequent development of the method [27]. Several reasons motivated us to take a closer look at Wilkinson's work. Among all contributions to inverse iteration, Wilkinson's are by far the mos... |

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Citation Context ... of ideas over several papers makes it difficult to realise what Wilkinson 2 [11, x5.4], [41, xx4.6-9], [40], [46, xIV.1.3] [62, x3] 3 [10, x3], [11, x5.9], [38, x2, x3], [45, x4] 4 [10, 29, 43, 8] 5 =-=[44, 45, 57, 62, 63]-=- 3 has accomplished. Therefore we decided to compile and order his main results and to set out his ideas. Wilkinson's numerical intuition provided him with many insights and empirical observations for... |

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Citation Context ...very good eigenvector ' [62, p 372]. Little or no work should be involved in determining these starting vectors. Here we assume exact arithmetic. The finite precision case is discussed in x6.2. Varah =-=[54]-=- and Wilkinson 13 showed that there exists at least one canonical vector that, when used as a starting vector x 0 , gives an iterate z 1 of almost maximal norm and hence an almost minimal residual r 1... |

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Citation Context ..., x1.2]. Although N is not unique, kNkF is. Here we are interested in the two-norm, kNk, which is not unique. But since Frobenius and two norms are related by [17, x2.2] 1 p n kNkFskNkskNkF ; 21 e.g. =-=[15, 22, 35, 36]-=- 21 we are content to know that kNk is at most p n away from a unique bound. We measure the departure of A \Gamma I from normality by j j kNk k( \GammasI) \Gamma1 k = kNk=ffl: This is a relative measu... |

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