On Extensions of the Power Method for Normal Operators

In a recent work by Sidi and Bridger some old and some new extensions of the power method have been considered, and some of these extensions have been shown to produce estimates of several dominant eigenvalues of an arbitrary square matrix. In the present work we continue the analysis of two versions of one of these extensions, called the MPE extension, as they are applied to normal matrices. We show that the convergence rate of these methods for normal matrices is twice that for nonnormal matrices. We also give precise asymptotic bounds on the errors of the estimates obtained for the eigenvalues. Further deflation-type extensions of the power method for normal matrices are suggested and analyzed for their convergence. All the results are stated and proved in the general setting of inner-product spaces. 1. I N T R O D U C T I O N In a recent work by Sidi and Bridger [1] some old and some new extensions of the power method have been considered. It has been shown for some of these extensions that they enable one to estimate several dominant eigenvalues of an arbitrary square matrix. In the present work we continue this analysis for one of these extensions, namely, the minimal polynomial extrapolation (MPE) extension, as it is employed in estimating the dominant eigenvalues of a normal matrix. (MPE is a method used in accelerating the convergence of vector sequences. For information and references pertaining to MPE and other similar methods, see [1] and the references therein.) To proceed, we give a brief description of two versions of the MPE extension of the power method in the notation and general setting of [1]. LINEAR ALGEBRA AND ITS APPLICATIONS 120:207-224 (1989) 207 © Elsevier Science Publishing Co., Inc., 1989 655 Avenue of the Americas, New York, NY 10010 0024-3795/89/$3.50 208 AVRAM SIDI Let B be an inner-product space over the field of complex numbers, and let (x, y) be the inner product associated with B. The homogeneity property of the inner product is such that for a and ]3 complex numbers, and x and y vectors in B, (ax, By)=~[J(x,y). Let also Ilxll--(x~-~,x~ be the norm associated with B. Let x 0, xl, x 2 . . . . , be a sequence of vectors in B, and assume that x m has an asymptotic expansion of the form ~ m m X m ~ h j a S m o oo, (1.1) j = l i where h~4:0, i = 1 , 2 . . . . , and h~4:h/ if i ~ j , (1.2) and Ihxl ~ Ih21 ~ " ' " , (1.3) and the vectors Yii, 0 ~< i ~< pl, j = I, 2 . . . . . are linearly independent. Here ( 7 ) are the binomial,coefficients. We further assume that there can be only a finite number of h i s having the same modulus. We agree to order the h i so that if IXjl -Ihi+11 for some j, then p/>/Pi+I . The interpretation of (1.1) is that for any positive integer N there exist a positive constant K and a positive integer m 0 that depend only on N, such that for every m >/m o, Xm j = l Yii~ i }] / ~ KlhNlmmp~'" (1.4) Special cases of vector sequences X m, m = 0, 1,2 . . . . . of the form described above arise naturally from the iterative process xj+l= Axi, j = 0, I . . . . . where A is an M × M complex matrix and x 0 is an arbitrary M-dimensional complex column vector. For this case the h i are nonzero eigenvalues of the matrix A, and the vectors Yt~' 0 ~< i ~< p j, are some linearly independent combinations of the eigenvectors and principal vectors corresponding to h i. For these special cases (1.1) takes on a simpler form, in the sense that the infinite sum on the fight-hand side of (1.1) is replaced by a finite sum and (1.1) reduces from being an asymptotic equivalence to being an equality. For more details on this see [1, Sections 2 and 6]. EXTENSIONS OF THE POWER METHOD 209 The extensions of the power method that have been suggested in [1] are all meant to produce approximations to )~t, ~2 . . . . . and they are based on knowledge of the vectors x i only. In all the extensions the approximations to the largest h j are obtained as the zeros of a polynomial k = E c:n ', (2.5) i = 0 whose coefficients are determined from the xi. All the extensions differ from one another in the way the c~ n'k) are determined. For the MPE extension c~ "'k) = c i are determined from either k ~_, ( x , + , , x , + j ) c j = O , O < ~ i < ~ k 1 , Ck =l , (1.6a) j=0