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A Power Method for Computing Square Roots of Complex Matrices

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A Power Method for Computing Square Roots of Complex Matrices JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 213, 393]405Ž. 1997 ARTICLE NO. AY975517 A Power Method for Computing Square Roots of Complex Matrices Mohammed A. Hasan Department of Electrical Engineering, Colorado State Uni¨ersity, Fort Collins, Colorado 80523 View metadata, citation and similar papers at core.ac.uk brought to you by CORE Submitted by Harlan W. Stech provided by Elsevier - Publisher Connector Received August 22, 1995 In this paper higher order convergent methods for computing square roots of nonsingular complex matrices are derived. These methods are globally convergent and are based on eigenvalue shifting and powering. Specifically, it is shown for each positive integer r G 2, a convergent method of order r can be developed. These algorithms can be used to compute square roots of general nonsingular complex matrices such as computing square roots of matrices with negative eigenvalues. Q 1997 Academic Press 1. INTRODUCTION m=m A square root of a complex matrix A g C is defined to be any m=m 2 matrix B g C such that B s A, where C is the field of complex numbers. If all eigenvalues of an m = m matrix A are distinct, then the 2 m matrix equation X s A generally has exactly 2 solutions. This follows from the fact that A is diagonalizable, i.e., there exists a similarity matrix y1 Ž. Usuch that A s UDU , where D s diag l1,...,lm and thus B s y1''ŽŽ .ii1 Ž .m . UDU, where D s diag y1 ''l1,..., y1 lmk, and i s 0or1 for k s 1, 2, . , m. However, if A has multiple eigenvalues, the number of m solutions will be different from 2 as shown next. Let m s 2, then without loss of generality, we can assume that A l220or A l1. s 0l22s 0l 2 cosŽ.u sin Ž.u Assume that A l I, then the family l 0 F u - 2p forms an s ½5sinŽ.u ycos Ž.u infinite set of square roots of A. On the other hand, if A l2 1 , then s 0 l2 A has only two square roots given by"l "1r2 l provided that l / 0. 0 "l Unlike the square roots of complex numbers, square roots of complex 393 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved. 394 MOHAMMED A. HASAN matrices may not exist. For example, when l s 0 in the last matrix no square root exists. From this observation, it is obvious that for 2 = 2 2 matrices, the equation B s A / 0 has a solution if and only if A has a nonzero eigenvalue. 2 To understand the structure of solutions of the equation B s A for bb m 2, let B 11 12 , and A aa11 12 such that B 2 A. Then we s s bb21 22 s aa21 22 s have the following four equations bbi11jiqbb22jijsa for i, j s 1, 2 which are equivalent to F s 0, where bb bb a ¡11 11q 12 21y 11 bb11 12qbb 12 22ya 12 FbŽ.11,b 12,b 21,b 22 s~ Ž1. bb21 11qbb 22 21ya 21 ¢bb21 12qbb 22 22ya 22 . The Jacobian of this system can be shown to be 2bb11 21 b12 0 ­ FbŽ.11,b 12,b 21,b 22 bb12 11qb 220 b 12 J ss . ­Ž.b11,b 12,b 21,b 22 b21 0b11qbb 22 21 0 bb21 12 2b22 Ž.2Ž. It can be verified that <<J s 4 b11q bbb 22 11 22ybb 12 21 s 4 Trace2Ž.BB<<. Here the notation <<J denotes the determinant of J, and Ž. Ým << Trace B s is1 bii. Since A is nonsingular, it follows that B / 0 and therefore J is nonsingular if and only if b11q b 22 / 0. Now assume that A is nonsingular and let bb00 B 11 12 0 s 00 bb21 22 2 0 0 be a solution of the equation B s A such that b11q b 22 / 0. Since J is 0000 nonsingular at Žb11, b 12, b 21, b 22., it follows from the implicit function theorem that B0 is the only solution. From the eigendecomposition of A indicated before, one can see that there are at least four square roots of A. The implicit function theorem guarantees exactly four square roots with nonzero traces. These square roots of A which have nonzero traces are referred to as functions of A wx1 . Essentially, B is a function of A if B can be expressed as a polynomial in A. Ž. 2 Now if b11q b 22 s 0, then it follows from 1 that a11q a 22 s l , 2 a12s a 21 s 0, i.e., A is diagonal of the form l I. In this case the equation SQUARE ROOTS OF COMPLEX MATRICES 395 2 B s A has a two-dimensional family of solutions given by 2 "'l y rs r r, s g C. ½5s.'l2 rs y Note that when r s s s l sinŽ.u , we get the one-parameter family l cosŽ.u sin Ž.u 0 u - 2p which is described before. ½5sinŽ.u ycos Ž.u F The following result provides conditions on the eigenstructure of A which ensure the existence of square roots which are functions of A. PROPOSITION 11.wx Let A be nonsingular and its p elementary di¨isors be coprime, that is, each eigen¨alue appears in only one Jordan block. Then A has precisely 2 p square roots, each of which is a function of A. Several computational methods of square roots of complex matrices have been reported in the literature. Inwx 2 , the Newton-Raphson method was used for computing the principal square root of a complex matrix. An accelerated algorithm for computing the positive definite square root of a positive definite matrix was presented inwx 3 . A matrix continued fraction method was presented inwx 4 . The matrix sign algorithm was developed in wx5 . A Schur method for computing square roots was developed in wx 6 . Fast stable methods for computing square roots were also presented inwx 7, 8 . It is noted in almost all of the above methods either a linear or quadratic convergence can be obtained. In this paper, higher order conver- gent methods of order r G 2 will be derived. The essence of these methods is a process whereby a sequence of matrices which in the limit converges to a square root of A is generated. This process involves creating gaps between the magnitudes of eigenvalues of different square roots of A so that for sufficiently high powers the eigenvalues will become decoupled. This is similar in principle to well-known methods such as those of Graeffe, Bernoulli, and the qd algorithm for solving polynomial equations in that these methods are based on eigenvalue powering. For a survey of some of these methods the reader is referred towx 9, 10 and the references therein. Let S be a set of commutative and thus simultaneously diagonalizable matrices. In the sequel, the notation liŽ.X denotes the ith eigenvalue of the square matrix X g S relative to a fixed similarity matrix which diagonalizes the set S. The notation s Ž.A denotes the set of eigenvalues of A. The symbol R is used to denote the set of real numbers and 55A denotes any vector norm of the matrix A. 396 MOHAMMED A. HASAN 2. DERIVATION OF THE MAIN RESULTS In the next theorem we will generate a sequence which converges to a square root of a square matrix. m=m THEOREM 2. Let A g C be a nonsingular matrix. Let r be a positi¨e integer such that r G 2 and define Akk and B recursi¨ely as follow. Let r r A ABAry2l2ll,2Ž. kq1sÝž/2l kk ls0 and r r B ABAry2 ly12lq1l.3Ž. kq1sÝž/2l1kk ls0 q Then there exists an a g C such that Bk is nonsingular for all sufficiently large y1 k. Set Xkkks B A , then the initial guess A0s aIm and B0s Im the sequence Xk con¨erges to a square root W of A. Moreo¨er, r y1 r Xkq1" W s BBXkq1kkŽ."W,4 Ž. i.e., if the sequence Xk con¨erges, it is rth order con¨ergent to W. Addition- ally, lim Ay1 Ar I and lim By1 B r I. k ª` kq1 kks ª`kq1ks 2 Proof. Let W be any square root of A, i.e., W s A and show by induction that r k Akk" BWsŽ.aI " W .5Ž. ClearlyŽ. 5 holds for k s 0. Assume thatŽ. 5 holds for the positive integer k. Then r r kq 1 r r Ž.ŽaI " W A " BW . ABAry2 l 2 ll s kks Ýž/2l kk ls0 r r " ABAWry2ly12lq1l Ýž/2l1kk ls0 q sAkq1"BWkq1 , where the last equality follows fromŽ. 2 and Ž. 3 . Hence Ž. 5 is true for the integer k q 1. This shows thatŽ. 5 is true for each nonnegative integer k. SQUARE ROOTS OF COMPLEX MATRICES 397 The nonsingularity of A implies that there exists an a g C such that Ž.Ž. Ž. Ž < ljjaI q W <<) l aI y W <, for j s 1,...,m. From 5 we have aI q .rk Ž.rk WsAkkqBW and aI y W s Akky BW. Solving the last two equations for Akkand B yields 1 rrkk AksÄ4Ž.Ž.aI q W q aI y W ,6 Ž.
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