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Block Power Method for Computing Solvents and Spectral Factors of Matrix Polynomials

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Block Power Method for Computing Solvents and Spectral Factors of Matrix Polynomials Comput. Math. Applic. Vol. 16, No. 9, pp. 683-699, 1988 009%4943/88 $3.00+0.00 Printed in Great Britain. All dights reserved Copyright © 1988 Pergamon Press plc BLOCK POWER METHOD FOR COMPUTING SOLVENTS AND SPECTRAL FACTORS OF MATRIX POLYNOMIALS J. S. H. TSAI,'~ L. S. SHmH and T. T. C. SHEN Department of Electrical Engineering, University of Houston, Houston, TX 77004, U.S.A. (Received 16 November 1987; in revisedform 9 February 1988) Communicated by E. Y. Rodin Abstract--This paper is concerned with the extension of the power method, used for finding the largest eigenvalue and associated ¢igenvector of a matrix, to its block form for computing the largest block eigenvalue and associated block eigenvector of a non-symmetric matrix. Based on the developed block power method, several algorithms are developed for solving the complete set of solvents and spectral factors of a matrix polynomial, without prior knowledge of the latent roots of the matrix polynomial. Moreover, when any right/left solvent of a matrix polynomial is given, the proposed method can be used to determine the corresponding left/right solvent such that both right and left solvents have the same eigenspectra. The matrix polynomial of interest must have distinct block solvents and a corresponding non-singular polynomial matrix. The established algorithms can be applied in the analysis and/or design of systems described by high-degree vector differential equations and/or matrix fraction descriptions. 1. INTRODUCTION There are numerous methods for computing the largest (scalar) eigenvahie and associated eigenvector of a matrix. The power method [1, 2] is one of the simplest procedures for computing the largest eigenvalue and associated eigenvector of a matrix, A of dimension n x n. This method in essence takes successively high powers of the vector resulting from the product of the normalized matrix and a non-trivial initial vector, and this vector converges to the eigenvector associated with the largest eigenvalue of the matrix. The convergence rate of the power method, in general, is very slow; however, it can be improved by an appropriate choice of a shift factor of the form, A - cln, where e is a constant scalar and In is an n x n identity matrix. By taking advantage of certain orthogonal relationships, efficient algorithms have been developed for computing the largest eigenvalue and the associated eigenvector of a symmetric matrix. Two such methods are the subspace iteration method [3], which can find several eigenvalues and the associated eigenvectors simultaneously, and the Lanczos method [4], which utilizes the best approximated eigenvector generated from the initial vectors to improve the computational aspects. It has been reported in Ref. [3] that significant progress has recently been made on the non-symmetric Lanczos algorithms; however, it has also been indicated [3] that much theoretical algorithmic development remains to be done. The purpose of this paper is to extend the (scalar) power method to its block form for use in the computation of the block eigenvalues and associated block eigenvectors of non-symmetric matrices. The block power method is based on the convergence of a block vector formed from the product of high powers of the normalized matrix and a non-trivial initial block vector to the block eigenvector associated with the largest block eigenvalue of the matrix. The largest block eigenvalue is also the largest solvent of the characteristic matrix polynomial of the matrix. This method is then utilized in developing various algorithms for determining the complete set of distinct solvents and spectral factors of a matrix polynomial whose corresponding polynomial matrix is non-singular. The paper is organized as follows. In Section 2, we review some definitions pertinent to the discussion of this paper. Next, the comprehensive proof and algorithms for finding the largest solvent of a monic matrix polynomial are developed in Section 3. Then, algorithms for finding the complete set of distinct solvents and spectral factors of a monic matrix polynomial are discussed in Section 4. Next, in Section 5, these algorithms are extended to cover general classes of matrix polynomials. Illustrative examples are given in Section 6, and the results are summarized in ~Now with the Department of Electrical Engineering, National Cheng-Kung University, Tainan, Taiwan, R.O.C. C.^.MW.A. ~6/~S 683 684 J.S.H. TSAI et al. Section 7. In this paper, we concentrate on the development of computational algorithms for finding the solvents and spectral factors of a matrix polynomial via the block power method rather than on the discussion of the computational complexity and numerical stability of the block power method. 2. REVIEW OF SOME DEFINITIONS OF A MATRIX POLYNOMIAL AND A POLYNOMIAL MATRIX For completeness of derivations presented in the latter part of this paper, we review pertinent definitions below. Definition 1 [5-I0] Consider an rth-degree mth-order monic matrix polynomial described by D(s) = i Ai s'-i, i=0 where Aie C ~ x % A0 = Ira, Im is an m x m identity matrix, and s is a complex variablel The left and right matrix polynomials of D(s) are defined by DL(X)= ~ Xr-'A, and DR(X)= ~ A,X'-', iffiO i=0 respectively, where X ~ C m× m. A left solvent L ~ C m×m is defined by DL(L) = i Lr-iAi =Om i=0 and a right solvent R ~ C ~ ×m is defined by DR(R) = ~ A,R~-'= Ore, i=0 where 0, is an m x m null matrix. Let 2i be a complex number such that det(D(2i))= 0; then, 2i is a latent root of D (s). Also, let D (s) have m linearly independent left latent vectors ql,. •., qm (right latent vectors pt ..... Pro) corresponding to latent roots 21 ..... 2~; then, Q-IAQ (PAP -I) is a left solvent L (right solvent R), where QT= (ql ..... qm) (P = (p~ ..... p,)) and A = diag(21 ..... 2m). A complete set of distinct left (right) solvents of D(s) is a set of m x m matrices L~(Ri) such that tT (Lj) = {2i, i = 1..... n } = a (D (2,)) tr (Rg) = {2,, i = 1..... n } = tr (D (2,)) jffi¿o (,oI ) and the block Vandermonde matrix VL(VR) is non-singular, where a(,) is the eigenspectrum of (,), and n = rm. The block Vandermonde matrices are defined as follows: Ii i L] L2...Lrl-l Imlrt I °°..''"Irtll VL ~ Im L2 L~... Li -I , VR ~ RI R2 ... R, . (la) fr f2r..,frr--I ,~--1 Rrg--I..°Rrr--I The complete set of distinct left solvents (Li) and right solvents (Ri) satisfy the following matrix equations, respectively: Lj+Ljr ,-I AI+L~-2A2+...+LjA,_t+A,=Om, j=l,2,., ., r, (lb) and R~+AiR~-I+A2R~-2+ ''' +Ar_IR/+A,=Om, j=1,2 ..... r. (lc) The block power method and matrix polynomials 685 The spectral factorization of the matrix polynomial D(s) can be described by D(s) = (st. - SO(st. - &) ..... (st. - S,), (ld) where S~ e C" ×'. The relationships among the left solvents (Lj), the right solvents (Rj) and the spectral factors (Sj) of D(s) can be found in Ref. [10]. For a general matrix polynomial, S~ # Lj, S~ # Rj, L; # Rj, for any i and j; except St = Lj and S, = Rj, for anyj. When A0 in D (s) is an identity matrix or a non-singular matrix, the matrix polynomial, D(s), is said to be monic or regular, respectively. Also, when A, in D(s) is an identity matrix or a non-singular matrix, the matrix polynomial, D(s), is referred to as co-monic or co-regular [6], respectively. • Definition 2 [11, 12] Define D[s]=[dij(s)], where dij(s) is a (scalar) polynomial in s, located at the ith row and the jth column in D Is]. D [s] is called a non-singular polynomial matrix when det(D[s])= det(D(s)) # 0, where the matrix polynomial D (s) has been defined in Definition 1. D [s] is said to be row-reduced (column-reduced) if the highest-row-degree coefficient matrix, denoted as Dhr (the highest-column-degree coefficient matrix, denoted as Dhc), is non-singular. In addition, it is defined as co-row-reduced (co-column-reduced) if the lowest-row-degree coefficient matrix, denoted as Du (the lowest-column-degree coefficient matrix, denoted as Dtc), is non-singular. • It has been shown [11, 12] that for a non-singular m x m polynomial matrix, /5Is], there exists a non-singular left polynomial matrix, P[s] (right polynomial matrix, Q[s]), such that L3 [s] = P [s]/) Is] (/5 Is] =/~ [s] Q Is]), where L~ [s] (/5 Is]) is either a row-reduced (column-reduced) or a co-row-reduced (co-column-reduced) polynomial matrix. P[s] (Q [s]) is a unimodular matrix if det(P[s])(det(Q Is])) is a constant. Definition 3 Define the eigenspectra of Gi6 C "×" and Gj~ C "×" as a(Gl)= {2~,1 for 1 = 1, 2 ..... m} and a (G2) = {2j.i for 1 = 1, 2 ..... m }, respectively. Also, define the absolute values of the eigenspectra of G, and Gj as I,~,,ll and I,~y,ll, for 1 = 1, 2,... ,m, respectively. In addition, la(G/)l > I~(Gj)l denotes {IA,,,I > IAj,A for/= 1,2,...,m}. • 3. BLOCK POWER METHOD FOR SOLVING THE SOLVENTS AND SPECTRAL FACTORS OF A MATRIX POLYNOMIAL Various computational algorithms [7, 8, 13-17] are available for finding the solvents and spectral factors of a matrix polynomial, or a ;t-matrix. A popular approach [8, 13, 14] is the use of the eigenvalues and eigenvectors of a matrix (represented in a bottom-type block companion form and having the g-matrix of interest as the characteristic matrix polynomial) to construct the solvents of the A-matrix based upon the definitions of solvents.
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