A Power Method for Computing Square Roots of Complex Matrices

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1 JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 13, ARTICLE NO. AY A Powe Method fo Computing Squae Roots of Complex Matices Mohammed A. Hasan Depatment of Electical Engineeing, Coloado State Uniesity, Fot Collins, Coloado 853 Submitted by Halan W. Stech Received August, 1995 In this pape highe ode convegent methods fo computing squae oots of nonsingula complex matices ae deived. These methods ae globally convegent and ae based on eigenvalue shifting and poweing. Specifically, it is shown fo each positive intege, a convegent method of ode can be developed. These algoithms can be used to compute squae oots of geneal nonsingula complex matices such as computing squae oots of matices with negative eigenvalues Academic Pess 1. INTRODUCTION A squae oot of a complex matix A C m m is defined to be any matix B C m m such that B A, whee C is the field of complex numbes. If all eigenvalues of an m m matix A ae distinct, then the matix equation X A geneally has exactly m solutions. This follows fom the fact that A is diagonalizable, i.e., thee exists a similaity matix 1 U such that A UDU, whee D diagž,...,. 1 m and thus B 1 i 1 i U DU, whee D diagžž 1.,..., Ž 1. m. 1 m, and i o1 fo 1,,..., m. Howeve, if A has multiple eigenvalues, the numbe of solutions will be diffeent fom m as shown next. Let m, then without 1 loss of geneality, we can assume that A o A. ½ 5 cos Ž. sinž. Assume that A I, then the family foms an sin Ž. cosž. infinite set of squae oots of A. On the othe hand, if A 1, then 1 A has only two squae oots given by povided that. Unlie the squae oots of complex numbes, squae oots of complex X97 $5. Copyight 1997 by Academic Pess All ights of epoduction in any fom eseved.

2 394 MOHAMMED A. HASAN matices may not exist. Fo example, when in the last matix no squae oot exists. Fom this obsevation, it is obvious that fo matices, the equation B A has a solution if and only if A has a nonzeo eigenvalue. To undestand the stuctue of solutions of the equation B A fo b11 b1 a11 a1 m, let B, and A such that B A. Then we b b a1 a 1 have the following fou equations bi1b1jbibjaij fo i, j 1, which ae equivalent to F, whee b 11 b 11 b 1 b 1 a 11 b11b1 b1 b a 1 FŽ b 11, b 1, b 1, b. Ž 1. b1b11 b b1 a1 b b b b a. The Jacobian of this system can be shown to be 1 1 b11 b1 b1 FŽ b 11, b 1, b 1, b. b1 b11 b b1 J. Ž b 11, b 1, b 1, b. b1 b11 b b1 b1 b1 b It can be veified that J 4Ž b b. Ž b b b b Tace Ž B.B. Hee the notation J denotes the deteminant of J, and m Tace B Ý b. Since A is nonsingula, it follows that B i1 ii and theefoe J is nonsingula if and only if b11 b. Now assume that A is nonsingula and let b b 11 1 b1 b B be a solution of the equation B A such that b11 b. Since J is Ž nonsingula at b, b, b, b , it follows fom the implicit function theoem that B is the only solution. Fom the eigendecomposition of A indicated befoe, one can see that thee ae at least fou squae oots of A. The implicit function theoem guaantees exactly fou squae oots with nonzeo taces. These squae oots of A which have nonzeo taces ae efeed to as functions of A 1. Essentially, B is a function of A if B can be expessed as a polynomial in A. Ž. Now if b11 b, then it follows fom 1 that a11 a, a1 a1, i.e., A is diagonal of the fom I. In this case the equation

3 SQUARE ROOTS OF COMPLEX MATRICES 395 B A has a two-dimensional family of solutions given by ½ s 5 s s, s C. Note that when s sin, we get the one-paamete family cos Ž. sinž. ½ 5 sin cos which is descibed befoe. The following esult povides conditions on the eigenstuctue of which ensue the existence of squae oots which ae functions of A. PROPOSITION 1 1. Let A be nonsingula and its p elementay diisos be copime, that is, each eigenalue appeas in only one Jodan bloc. Then A has pecisely p squae oots, each of which is a function of A. Seveal computational methods of squae oots of complex matices have been epoted in the liteatue. In, the Newton-Raphson method was used fo computing the pincipal squae oot of a complex matix. An acceleated algoithm fo computing the positive definite squae oot of a positive definite matix was pesented in 3. A matix continued faction method was pesented in 4. The matix sign algoithm was developed in 5. A Schu method fo computing squae oots was developed in 6. Fast stable methods fo computing squae oots wee also pesented in 7, 8. It is noted in almost all of the above methods eithe a linea o quadatic convegence can be obtained. In this pape, highe ode convegent methods of ode will be deived. The essence of these methods is a pocess wheeby a sequence of matices which in the limit conveges to a squae oot of A is geneated. This pocess involves ceating gaps between the magnitudes of eigenvalues of diffeent squae oots of A so that fo sufficiently high powes the eigenvalues will become decoupled. This is simila in pinciple to well-nown methods such as those of Gaeffe, Benoulli, and the qd algoithm fo solving polynomial equations in that these methods ae based on eigenvalue poweing. Fo a suvey of some of these methods the eade is efeed to 9, 1 and the efeences theein. Let S be a set of commutative and thus simultaneously diagonalizable matices. In the sequel, the notation Ž X. i denotes the ith eigenvalue of the squae matix X S elative to a fixed similaity matix which diagonalizes the set S. The notation Ž A. denotes the set of eigenvalues of A. The symbol R is used to denote the set of eal numbes and A denotes any vecto nom of the matix A. A

4 396 MOHAMMED A. HASAN. DERIVATION OF THE MAIN RESULTS In the next theoem we will geneate a sequence which conveges to a squae oot of a squae matix. THEOREM. Let A C m m be a nonsingula matix. Let be a positie intege such that and define A and B ecusiely as follow. Let and ž / l l l 1 Ý l l A A B A, ž / l1 l1 l 1 Ý l1 l B A B A. Ž 3. Then thee exists an a C such that B is nonsingula fo all sufficiently lage. Set X B 1 A, then the initial guess A aim and B Im the sequence X coneges to a squae oot W of A. Moeoe, 1 X1 W B1B XW, 4 i.e., if the sequence X ally, coneges, it is th ode conegent to W. Addition lim A 1 A I and lim B 1 B I. 1 1 Poof. Let W be any squae oot of A, i.e., W A and show by induction that A BW ai W. 5 Ž. Ž. Clealy 5 holds fo. Assume that 5 holds fo the positive intege. Then ž / ž l1/ 1 l l l l l Ž ai W. Ž A BW. Ý A B A A1B1W, Ý l1 l1 l l A B A W whee the last equality follows fom Ž. and Ž. 3. Hence Ž. 5 is tue fo the intege 1. This shows that Ž. 5 is tue fo each nonnegative intege.

5 SQUARE ROOTS OF COMPLEX MATRICES 397 The nonsingulaity of A implies that thee exists an a C such that Ž ai W. Ž ai W., fo j 1,...,m. Fom Ž. 5 we have Ž j j ai W. A BW and Ž ai W. A BW. Solving the last two equations fo A and B yields and 1 4 A Ž ai W. Ž ai W., Ž B Ž ai W. Ž ai W. W. Ž 7. Note that the ode of matix multiplication is immateial in this case since the choice X 4 ai implies that the elements of the sequence X 1 commute with each othe. It should also be noted that fo sufficiently lage both A and B ae invetible since Ž ai W. Ž ai W. j j, j Ž. 1 1,..., m. Thus multiplying both sides of 5 by B yields ½ ½ B A WB aiw W I ai W aiw W I aiw aiw aiw aiw. Since Ž ai W. Ž ai W. i i, fo i 1,...,m, it follows fom the last equation that 1 1 lim B A W lim B aiw W. Theefoe, lim B 1 A W. To pove Ž. 4 we have fom the elation A 1 B1W ABW that o equivalently B1A1WB1B B AW, 1 X1 W B1BŽ XW.. Finally, the last conclusion follows fom B1BŽ aiw. Ž aiw. 4 Ž aiw. Ž aiw ½žI Ž Ž ai W. Ž aiw.. 5 as. 1 ½ž Ž. / I ai W aiw I Q.E.D.

6 398 MOHAMMED A. HASAN An analogous algoithm to that of Theoem can be obtained by setting X B 1 A as shown next. THEOREM 3. Let A be a nonsingula matix and let be a positie intege such that. Let Ýž / l ž l1/ l l C X A, Ž 8. l Ý l1 l D X A, Ž 9. l and set X1 D 1 C. Then thee exists an initial matix X ai C m m, a C fo which the sequence X 4 1coneges to a squae oot W of A. Moeoe, 1 X1 W D XW, 1 i.e., if the sequence X coneges it is th ode conegent to W. Poof. B 1 A. Theoem 3 follows diectly fom Theoem by setting X Q.E.D. In Theoems and 3 and if, quadatically convegent algoithms ae obtained as follows. COROLLARY 4. Let A be an mm nonsingula complex matix and define the following sequence A1 A B A, and B1 AB with AaI and B I. Then fo some a C, the sequence B 1 A coneges 1 quadatically to A. Moeoe, if we set X B A, then the iteation 1 1Ž X X X X A. 1 with X ai coneges quadatically to some 1 1Ž. A and X A X X A Ž Rema. The iteation X X X X A. 1 of the above coollay is exactly the Newton iteation fo solving X A. Seveal vaiants of Newton s method and thei implementations wee pesented in, whee the matix A is assumed diagonalizable. Note that the algoithm of Coollay 4 and the Newton method ae equivalent. Howeve, the deivation of the above algoithm lends itself to the development of othe methods whose convegence is of any pescibed ode. A cubically convegent algoithm can be deived by setting 3 in Theoems and 3 as in the next esult. COROLLARY 5. Let A be an mm nonsingula complex matix and define the following sequence A1 A 3 3B A A and B1 3 A B B 3 A, with A ai and B I. Then fo some a C, the sequence B 1 A

7 SQUARE ROOTS OF COMPLEX MATRICES 399 coneges cubically to a squae oot of A. Moeoe, if we set X B 1 A Ž. 1 Ž then the sequence satisfies the ecusion X1 X 3X A X A. which with the initial guess X ai coneges cubically to A and Ž. 1Ž. 3 X A3X A X A ANALYSIS OF CONVERGENCE To analyze the convegence of the sequence X 4 1we fomulate the iteation in Theoem as a fixed point iteation as established in the next esult. THEOREM 6. Assume thee exists an a C such that Ž i ai A. Ž ai A. i 1, fo i 1,...,m. Then the sequence X defined in Theoem 3 is th ode conegent, i.e., Ž. X A O X A, 1 o equialently, thee exists a constant S such that X ASX A. 1 mm Poof. Let Z C be any squae matix of A and define CZ l l Ý Z A and DZ Ý Z l1 A. l Then l ž/ l lž l1/ Ž. C Z D Z A Z A. 1 mm mm Define Z DZ CZ. Then : C C and 1 l1 l l l ½ Ýž l1/ 5 ½ Ýž l/ 5 l l Ž. Ž. Ž. A A A A A ½ A A A A. ½ 5 Theefoe A and A ae fixed points fo the function. Now, the function can be witten as Ž. 1 Ž. 1 Z A D Z C Z D Z A A D Z Z A, 1 hence Z A DZ Z A. Note that when Z is a scala it can be shown d 1 1 Ž Z. DŽ Z. Ž Z A.. dz

8 4 MOHAMMED A. HASAN Ž l Theefoe if A, then and d dz l. Ž., fo l 1,,..., 1. It follows that thee is an R and a neighbohood, mm S Z C Z R 4 R,of and a constant K,K1 such that Ž Z. Ž. KZ mm fo all Z S Z C Z R4 Ž R i.e., is a contactive mapping in SR Ž... Hence fo any X SR Ž. the geneated sequence X Ž X., 1,, has the popeties X S Ž. and X 1 R K X fo 1,,. This shows that the iteation X1 Ž X. conveges to fo any initial guess X in SRŽ. fo which X AAX. To pove the last conclusion of the theoem, we have Ž. C D A X A, Ž 11. ž / l l l l whee C C X Ý X A, and D D X ž / l1 l 1 Ýl X A. Since D A A is nonsingula, D is l 1 nonsingula fo all sufficiently lage. It follows that thee exists a 1 constant S such that D S fo all sufficiently lage. Theefoe X ASX A 1. This poves the th ode convegence of X. Q.E.D. 4. CONTINUED FRACTION EXPANSION Thee ae many fomulas in the liteatue that appoximate squae oots of numbes and matices using continued faction expansion 4. In this section, we expess the esults of Theoem 3 in continued faction fom. THEOREM 7. Let A be a nonsingula matix and let W be a squae oot of A. Then fo each a C fo which Ž ai W. Ž ai W. i i, i 1,...,m,W has the epesentation W ai Ž A a I. ½ ai Ž A a I ai Ž A ai. ai. Ž 1.

9 SQUARE ROOTS OF COMPLEX MATRICES 41 It can easily be veified that the th ode tuncation of the continued 1 faction of Ž 1. yields DaI CaI, whee CaI Ý a l A, l and ž / l ž/ l l1 l l l 1 DaI Ý a A. When A is a scala, Fomula 1 educes to A a A a Ž. a Aa a Aa a, Ž 13. which in the case a 1 becomes that of 4. Note that when A is positive, Ž 13. conveges fo any a C with nonzeo eal pat. The fee paamete a povides some flexibility in choosing the initial guess in that the close a is to A the moe acceleated is the convegence. Rema on the Choice of the Initial Guesses. In this ema we povide special cases whee conditions on the initial matix X can be imposed to ensue convegence. Assume that all eigenvalues of A ae not on the negative eal line. Then each eigenvalue of A has nonzeo eal pat. The initial guess A ai with a Ž a. foces the sequence X to convege to a squae oot of A with eigenvalues having positive Ž negative. eal pat. Howeve, if some of the eigenvalues of A ae on the imaginay axis, then the choice X Ž a ib. I, whee a and b, will lead to a sequence which conveges to a squae oot of A having eigenvalues with nonnegative eal pats. These obsevations show in paticula that if A is positive Ž o negative definite. then the methods of Theoems and 3 convege fo any initial guess X of the fom X ai, a R Ž ia R., whee i 1. This is an impovement ove the algoithm in which is applicable only to matices which have no negative eigenvalue. 5. COMPUTATIONAL RESULTS To demonstate the pefomance of the algoithms poposed in this pape we conside in this section some examples to show the behavio of some of these methods in finite-pecision aithmetic. These computations wee caied out on appoximately seven decimal digit accuacy. Fo the pupose of compaison we will apply Theoem 3 to fou examples. The notation X, will denote the computed squae oot using iteations and th ode method. The eo is measued in the Fobenious nom X, A. F

10 4 EXAMPLE 1. MOHAMMED A. HASAN Conside the complex 3 3 matix 8 15i 1 3i 4 9i i 59i 13i A i 13i 815i Applying five iteations of the algoithm of Theoem 3 with, X 1i I, yields i i X5, i i i i i i i, which agees with the exact squae oot to five decimal places, i.e., Ž 6 X A O 1. 5, F. Compaable accuacy can also be achieved using only fou iteations with a thid ode method with the same initial guess Ž 6 in which case we obtain X A O 1.. 4, 3 EXAMPLE. Conside the 4 4 matix F A The eigenvalues of this matix ae Ž A. 1,, 5, 14 and -nom condition numbe Ž A. 1. When X I and 4 iteations ae used with 3, we obtain X 4, , Ž 6. Ž 3 and X A F O 1. We also note that X A F O 1. 4, 3 4, Ž 7 while X A O 1.. 5, F

11 SQUARE ROOTS OF COMPLEX MATRICES 43 EXAMPLE 3. Conside the 4 4 nea singula matix A The eigenvalues of this matix ae Ž A..3, 3.3, 1.97 i 4. Applying the iteation of Theoem 3 with 3 and X Ž 1 i. I we obtain X 7, E E E , Ž 4 with eo X A O 1. 7, 3 F. Howeve, when, simila accuacy can be obtained with the same X as X 6, E E E , Ž 4 with eo measued in Fobenious nom as X A O 1.. EXAMPLE 4. Conside the 3 3 matix 1 1 A 1 1, which has the eigenvalues A 1, 1,, i.e., this matix has a negative eigenvalue. Thus the initial matix X ai should be chosen so that 6, F

12 44 MOHAMMED A. HASAN the imaginay pat of a is nonzeo. Using X 1 i I, we obtain i i X9, i i i i i i i, Ž 1 with eo X A O 1. 5, F. If a method of ode thee is used, Ž 8 then X A O 1. 3, 3 F. In summay, some squae oots of nonsingula complex matices can be computed using only initial matices of the fom X Ž a ib. I. The case b is equied fo matices having negative eigenvalues. 6. CONCLUSION A new set of algoithms fo computing squae oots of nonsingula complex matices was developed. Given any positive intege we pesented a systematic way of deiving an th ode convegent squae oot algoithm. The convegence and its speed ae lagely affected by the atios R X A 4 X A 4 i i i 1, i 1,...,m, which ae ob- viously dependent on the initial guess X. Thus X seves as a fee paamete fo which a faste convegence can be achieved by choosing X so that the R i s ae close to zeo. It should be obseved that the conclusion of Theoem 3 is still valid if X ai is eplaced by any othe m m matix povided that X and A commute. Fo, this tech- nique becomes the Newton method fo solving the equation X A. One shotcoming of these algoithms is that they cannot be applied to compute all squae oots of A using only initial guesses of the fom X ai, a C. In ode to obtain all squae oots, a matix sign function of all squae oots is to be computed, i.e., fo each squae oot A, a matix S such that S I and S A AS must be geneated and then use S as an initial guess in Theoems o 3. Finally, this wo can be genealized to compute the nth oots of complex matices. This is the subject of a pape submitted fo publication 11. ACKNOWLEDGMENT The autho thans the efeees fo thei helpful emas and suggestions which impoved the quality of this wo.

13 SQUARE ROOTS OF COMPLEX MATRICES 45 REFERENCES 1. N. J. Higham, Computing eal squae oots of a eal matix, Linea Algeba Appl Ž 1987., N. J. Higham, Newton s method fo the matix squae oot, Math. Comp. 46, No. 174 Ž 1986., E. D. Denman, Roots of eal matices, Linea Algeba Appl. 36 Ž 1981., L. S. Shieh and N. Chahin, A compute-aided method fo the factoization of matix polynomials, Appl. Math. Comput. Ž 1976., E. D. Denman and A. N. Beaves, The matix sign function and computation of systems, Appl. Math. Comput. Ž 1976., A. Bjoc and S. Hammaling, A Schu method fo the squae oot of a matix, Linea Algeba Appl. 553 Ž 1983., W. D. Hosins and D. J. Walton, A fast method of computing the squae oot of a matix, IEEE Tans. Automat. Contol AC-3, No. 3 Ž 1978., W. D. Hosins and D. J. Walton, A fast, moe stable method fo computing the pth oot of positive definite matices, Linea Algeba Appl. 6 Ž 1979., P. Henici, Applied and Computational Complex Analysis, Vol. 1, Wiley, New Yo, A. S. Householde, The Numeical Teatment of a Single Nonlinea Equation, Mc- GawHill, New Yo, M. A. Hasan, Highe ode convegent algoithms fo computing nth oots of complex matices, submitted fo publication.