An iterative method to solve a nonlinear matrix equation

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1 Electroic Joural of Liear Algebra Volume 3 Volume 3: (206) Article A iterative method to solve a oliear matrix equatio Peg Jigjig Hua Uiversity, Chagsha 40082, PR Chia, jjpeg202@63.com Liao Apig Hua Uiversity, Chagsha 40082, PR Chia Peg Zheyu Guili Uiversity of Electroic Techology, Guili, yuzhep@63.com Follow this ad additioal works at: Part of the Numerical Aalysis ad Computatio Commos Recommeded Citatio Jigjig, Peg; Apig, Liao; ad Zheyu, Peg. (206), "A iterative method to solve a oliear matrix equatio", Electroic Joural of Liear Algebra, Volume 3, pp DOI: This Article is brought to you for free ad ope access by Wyomig Scholars Repository. It has bee accepted for iclusio i Electroic Joural of Liear Algebra by a authorized editor of Wyomig Scholars Repository. For more iformatio, please cotact scholcom@uwyo.edu.

2 AN ITERATIVE METHOD TO SOLVE A NONLINEAR MATRIX EQUATION JINGJING PENG, ANPING LIAO, AND ZHENYUN PENG Abstract. I this paper, a iterative method to solve oe kid of oliear matrix equatio is discussed. For each iitial matrix with some coditios, the matrix sequeces geerated by the iterative method are show to lie i a fixed ope ball. The matrix sequeces geerated by the iterative method are show to coverge to the oly solutio of the oliear matrix equatio i the fixed closed ball. I additio, the error estimate of the approximate solutio i the fixed closed ball, ad a umerical example to illustrate the covergece results are give. Key words. Noliear matrix equatio, Iterative method, Newto s iterative method, Covergece theorem. AMS subject classificatios. 5A24, 5A39, 65F30.. Itroductio. We cosider the oliear matrix equatio (.) X +A X A = Q, where Q is a m m positive defiite matrix, A is a arbitrary real m m matrix, ad is a positive iteger greater tha. The solutio of the matrix equatio (.) is a problem of practical importace. I may physical applicatios, oe must solvethe system of the liear equatio Mx = f, where the positive defiite matrix M arises from a fiite differece approximatio to a elliptic partial differetial operator (e.g. see [2]). As a example, let M = The M = M +diag(i X,0), where ( I A A Q M = ( X A A Q ). ). Received by the editors o March 24, 205. Accepted for publicatio o September 3, 206. Hadlig Editor: Brya L. Shader. Research was supported by Natioal Natural Sciece Foudatio of Chia (o. 2604, o. 3007). Hua Uiversity, Chagsha 40082, PR Chia (jjpeg202@63.com, liaoap@hu.c). Guili Uiversity of Electroic Techology, Guili 54004, PR Chia (yuzhep@63.com). 620

3 Iterative Methods to Solve a Noliear Matrix Equatio 62 If X is a positive solutio to (.), the the LU decompositio of M is ( ) ( )( ) X A I 0 X A A = Q A X, I 0 X the system Mx = f is trasformed to [ ( )( I 0 X A A X I 0 X ) ( I X ) ] x = f, ad the Sherma-Morriso-Woodbury formula [25] ca be applied to compute the solutio of Mx = f directly, or based o the iterative formula ( )( ) ( ) I 0 X A I X 0 A X x k+ = f x k, k = 0,,2,... I 0 X 0 0 to solve the system Mx = f. Aother example is i cotrol theory, see the refereces give i [9]. Several authors have studied the existece, the rate of covergece as well as the ecessary ad sufficiet coditios of the positive defiite solutios of (.) or similar matrix equatios. For example, the fixed poit iteratio methods to solve (.) have bee ivestigated i [6, 7, 4, 26]. The properties of the positive defiite solutios of (.) ad the perturbatio aalysis of the solutios have bee itroduced i [23, 27]. Some special cases of (.) have bee ivestigated. Fixed poit iteratio methods to solve (.) i cases = 2 ad Q = I have bee studied i [4, 28, 29, 36]. For the case = or cases = ad Q = I, fixed poit iteratio methods are cosidered i [, 4, 5,, 2, 5, 7, 9, 20, 22, 24, 32]. The iversio free variat of the basic fixed poit iteratio methods are cosidered i [0, 6, 38]. Cyclic reductio methods are cosidered i [8, 9, 32]. Properties of the positive defiite solutios of (.) ad perturbatio aalysis of the solutios have bee discussed i [20, 22, 24]. More geeral cases have also bee aalyzed [3, 8, 2, 30, 33, 34, 35, 37]. I this paper, we cosider Newto s iterative method to solve the oliear matrix equatio(.). Weshow,forthegivetheiitialmatrixX 0, thatthematrixsequeces {X k } geerated by the iterative method areicluded i a fixed ope ball B(X 0,). We also show that the {X k } geerated by the iterative method coverges to the oly solutio of the oliear matrix equatio (.) i the fixed closed ball B(X 0,). I additio, the error estimate of the approximate solutio i the fixed closed ball B(X 0,) is preseted, ad a umerical example to illustrate the covergece results is give. We use A F ad A to deote the Frobeius orm ad the spectral orm of the matrix A, respectively. The otatio R m ad H deotes, respectively, the set of all complex m matrices ad all Hermitia matrices. A stads for

4 622 J. Peg, A. Liao, ad Z. Peg the cojugate traspose of the matrix A. A B stads for the Kroecker product of the matrices A ad B. 2. Covergece of the iterative methods. I this sectio, we discuss Newto s iteratio method to solve the oliear matrix equatio (.), ad show that the iterates coverge to the oly solutio of the oliear matrix equatio (.) i a fixed closed ball. Let F(X) = X +A X A Q, the matrix fuctio F(X) is Frechet-differetiable at the osigular matrix X ad the Frechet-derivate is give by (2.) We have the followig lemmas. F X (E) = E A X i EX +i A. Lemma 2.. Let X be a osigular with X + A 2 <. The the liear operator F X is osigular, ad (F X ) X + A 2. Proof. The iequality F X(E) = E A X i EX +i A E A X i EX +i A E X + A 2 E ad the assumptio X + A 2 < imply that F X (E) = 0 if ad oly if E = 0, that is, the operatorf X is a ijectio. Sice F X is a operatoro the fiite dimesio liear space C, F X is a surjectio. Therefore, F X is osigular, ad (F X) = mi{ F X (H) / H : H 0} X + A 2. Lemma 2.2. For osigular matrices X ad Y, we have (2.2) + F X F Y A 2 ( X +2 i Y i ) X Y.

5 Iterative Methods to Solve a Noliear Matrix Equatio 623 Proof. For a arbitrary matrix E, we have (F X F Y )(E) = A X i EX +i A = + = = + = + Therefore, (2.2) holds. A X i EX +i A A X i EY +i A A Y i EY +i A A X i EY +i A A Y i EY +i A A X i E(X +i Y +i )A+ A (X i Y i )EY +i A i A X i E[ X ( i k) (X Y )Y k ]A i A [ X k (X Y )Y (i k) ]EY +i A i A X i E[ X ( i k+) (Y X)Y k ]A i A [ X k (Y X)Y (i k) ]EY +i A i A 2 ( X k+ Y k+ ) X Y E + A 2 X k+ Y k+ ) X Y E i ( i = A 2 ( X k+ Y k+ ) X Y E + A 2 ( k= i+ X k+ Y k+ ) X Y E = A 2 ( X k+ Y k+ ) X Y E + = A 2 ( X +2 i Y i ) X Y E.

6 624 J. Peg, A. Liao, ad Z. Peg The followig lemma from [3, 3] will be useful to prove the ext theoretical results. Lemma 2.3. Let A,B R ad assume that A is ivertible with A α. If A B β ad αβ <, the B is also ivertible, ad B α/( αβ). Applyig Newto s method to solve the oliear matrix equatio (.), we have (2.3) or equivaletly, (2.4) (2.5) X k+ = X k (F X k ) (F(X k )), k = 0,,2,... E k A X i k E kx +i k A = F(X k ), X k+ = X k +E k, k = 0,,2,... Lettig H(X) = (F X 0 ) (F(X)), we see that the Newto iterates for the matrix fuctio H(X) coicide with those of the matrix fuctio F(X) sice (2.6) X k+ X k = (H X k ) (H(X k )) = (F X k ) (F(X k )). (2.7) Lemma 2.4. Let X 0 be a osigular matrix such that holds, ad let The 0 < = (+)( X 0 A 2 + Q X 0 ) X (i) H(X 0 ) + ; 0 + A 2 < ( X B(X 0,) = {X X X 0 < }. (ii) H X H Y X Y for all X,Y B(X 0,); (iii) (H X ) X X 0 / for all X B(X 0,); ) /(+2) X0 (iv) H(X) H(Y) H Y (X Y) 2 X Y 2 for all X,Y B(X 0,). Proof. (i) By the defiitio of ad the estimate obtaied by Lemmas 2. ad 2.2, we have H(X 0 ) = (F X 0 ) (F(X 0 ))

7 Iterative Methods to Solve a Noliear Matrix Equatio 625 (F X 0 ) F(X 0 ) X0 + A 2.( X 0 A 2 + Q X 0 ) = X 0 A 2 + Q X 0 X0 = + A 2 +. (ii) Usig the estimates obtaied by Lemma 2. ad 2.2, we have H X H Y = (F X 0 ) F X (F X 0 ) F Y (F X 0 ) F X F Y) + X0 ( X +2 i Y i ) X Y. + A 2. A 2 Sice X X 0 <, Y X 0 < ad X0 < ( X ) /+2 <, we have by Lemma 2.3 that X ad Y are both osigular, ad X X 0 Y X X. Therefore, 0 H X H Y X A A 2.(+) X ( X0. X Y )+2 X0 + A 2.(+) X =. X0 2 2 ( X Y )+2. X 0 X Y. X, ( X0 A 2 + Q X 0 ) ( X 0 )+2. X Y (iii) Accordig to the defiitio of the matrix fuctio H(X), it is easy to get (H X 0 ) =. The estimate (ii) implies that H X H X 0 X X 0 < for all X B(X 0,). Thus, we have by Lemma 2.3 that (H X ) X X 0 / for all X B(X 0,). (iv) Usig the Newto-Leibiz formula ad (iii), we have H(X) H(Y) H Y (X Y) = (H( t)y +tx H Y )(X Y)dt 0 X Y 0 H ( t)y+tx H Y dt X Y 2 tdt = 2 X Y 2. Lemma 2.5. Assume that the iitial matrix X 0 satisfies (2.7). The the Newto s iterates X k, k 0, for the matrix fuctio H(X), ad hece for the matrix fuctio 0

8 626 J. Peg, A. Liao, ad Z. Peg F(X), belog to the ope ball B(X 0,), ad furthermore, X k X k hold for all k. 2 k (+), X k X 0 ( 2 k (+) ), (H X k ) 2 k (+), H(X k ) 2 2k (+) 2. Proof. First, let us check that the above estimates hold for k =. Clearly, the poit X = X 0 (H X 0 ) (H(X 0 )) = X 0 H(X 0 ) is well defied sice H X 0 is ivertible. Also ad by (iii) of Lemma 2.4, X X 0 = H(X 0 ) + < ( + ), (H X ) X X 0 / +. By defiitio of X, ad by (iv) of Lemma 2.4 agai, H(X ) = H(X ) H(X 0 ) H X 0 (X X 0 ) 2 X X 0 2 2(+) 2. Assume that the estimates hold for k =,2,...,m for some iteger m. The poit X m+ = X m (H X m ) (H(X m )) is thus well defied sice H X m is ivertible. Moreover, by the iductio hypothesis ad by estimates of (iii) of Lemma 2.4 (for the third ad fourth estimates), X m+ X m (H X m ) H(X m ) 2 m (+), X m+ X 0 X m X 0 + X m+ X m ( 2 m (+) )+ 2 m (+) = ( 2 m (+) ), (H X m+ ) X m+ X 0 / 2m (+), H(X m+ ) = H(X m+ ) H(X m ) H X m (X m+ X m ) 2 X m+ X m 2 2 2m+ (+) 2.

9 Iterative Methods to Solve a Noliear Matrix Equatio 627 Hece, the estimates also hold for k = m+. Lemma 2.6. Assume that the iitial matrix X 0 satisfies (2.7), the the Newto s iterates X k, k 0, coverges to a zero X of H(X), ad hece of F(X), which belogs to the closed ball B(X 0,). Moreover, for all k 0. X k X 2 k (+) Proof. The estimates X k X k, k, established i Lemma k (+) clearly imply that {X k } k= is a Cauchy sequece. Sice X k B(X 0,) B(X 0,) ad B(X 0,) is a complete metric space (as a closed subset of the Baach space), there exists X B(X 0,) such that X = lim k X k. Sice H(X k ) fuctio, 2 2k (+) 2, k, by Lemma 2.5, ad H(X) is a cotiuous H( X) = lim k H(X k) = 0. Hece, the poit X is a zero of F(X). Give itegers k ad l, we have, agai by Lemma 2.5, X k X k+l so that, for each k, l+p j=k X j+ X j j=k 2 j (+) = 2 k (+), X k X = lim X k X k+l k 2 k (+). Lemma 2.7. Assume that the iitial matrix X 0 satisfies (2.7), the the zero X of H(X), ad hece of F(X), i the closed ball B(X 0,) uique. Proof. We first show that, if Z B(X 0,) ad H( Z) = 0, the X k Z 2 k (+) for all k 0. Clearly, this is true if k = 0; so assume that this iequality holds for k =,2,...,m, for some iteger m 0. Notig that we ca write X m+ Z = X m (H X m ) (H(X m )) Z = (H X m ) (H( Z) H(X m ) H X m ( Z X m )),

10 628 J. Peg, A. Liao, ad Z. Peg we ifer from Lemma 2.4 ad 2.5, ad from iductio hypothesis that X m+ Z (H Xm ) 2 Z X m 2 Hece, the iequality X k Z which shows that Z = X. 2 k (+) lim X m Z = X Z = 0, m Lemmas ow imply the followig. 2 m (+). holds for all k. Cosequetly, Theorem 2.8. Assume that the iitial matrix X 0 satisfies (2.7). The the sequece {X k } defied by (2.3) is such that X k B(X 0,) for all k 0 ad coverges to a solutio X B(X 0,) of the oliear matrix equatio (.). Moreover, for all k 0, X k X 2 k (+), ad the poit X is the oly solutio of the oliear matrix equatio (.) i B(X 0,). 3. Numerical examples. I this sectio, we preset a umerical example to illustrate the covergece results of the Newto s method to solve the equatio (.). Our computatioal experimets were performed o a IBM ThikPad of mode T40 with 2.5 GHz ad 3.0 RAM. All tests were performed i the MATLAB 7. with a 32-bit Widows XP operatig system. The followig example we cosider the matrix equatio (.) with = 2, ad use LSQR M algorithm from [33] to solve the subproblem (2.4) of the Newto s method. For coveiece, we let A = A X k,a 2 = A X 2 k,b = X 2 k A,B 2 = X k A,C = F(X k). The the LSQR M algorithm to solve the subproblem (2.4), that is, ca be described as follows. E +A EB +A 2 EB 2 = C LSQR M:(Algorithm for solvig the matrix equatio (2.4) with = 2) () Iitializatio. Set iitial matrix E 0 H. Compute β = C E 0 A E 0 B A 2 E 0 B 2 F, U = (C E 0 A E 0 B A 2 E 0 B 2 )/β, α = U +A T U B T +AT 2 U B T 2 F, V = (U +A T U B T +A T 2U B T 2 )/α, W = V, φ = β, ρ = α. (2) Iteratio. For i =,2,..., util the stoppig criteria have bee met

11 Iterative Methods to Solve a Noliear Matrix Equatio 629 (a) β i+ = V i +A V i B +A 2 V i B 2 α i U i F,U i+ = (V i +A V i B +A 2 V i B 2 α i U i )/β i+, (b) α i+ = U i+ + A T U i+ B T + A T 2U i+ B T 2 β i+ V i F,V i+ = (U i+ + A T U i+b T +AT 2 U i+b T 2 β i+v i )/α i+. (c) ρ i = ( ρ 2 i + β2 i+ )/2, c i = ρ i /ρ i, s i = β i+ /ρ i, θ i+ = s i α i+, ρ i+ = c i α i+, φ i = c i φi, φ i+ = s i φi. (d) E i = E i +(φ i /ρ i )W i, (e) W i+ = V i+ (θ i+ /ρ i )W i. The stoppig criteria is used as φ k+ α k+ c k 0 0 here. Other stoppig criteria ca also be used, readers ca see [33] for details. Example 3.. Let = 2, ad give matrices A ad Q be as follows. A = , Q = By direct compute, we kow that Q is a positive defiite matrix, ad the followig estimates hold: = (+) Q A 2 Q + A 2 =.7778< ( Q 2 2 ) /(+2) Q = Hece, give matrices A ad Q satisfy the coditio of Theorem 2.8. Usig Newto s method ad iterate 4 steps, we have X 4 =

12 630 J. Peg, A. Liao, ad Z. Peg , with X A X 2 A Q F = , X 4 Q = < = Coclusios. I this paper, Newto s iterative method to solve the oliear matrix equatio X +A X A = Q is discussed. For the give iitial matrix X 0, the results that the matrix sequeces {X k } geerated by the iterative method are icluded i the fixed ope ball B(X 0,)(Lemma 2.5) ad that the matrix sequeces {X k } geerated by the iterative method coverges to the oly solutio of the matrix equatio X + A X A = Q i the fixed closed ball B(X 0,) (Lemmas 2.6 ad 2.7) are proved. I additio, the error estimate of the approximate solutio i the fixed closed ball B(X 0,) (Lemma 2.6), ad a umerical example to illustrate the covergece results are preseted. The advatage of Newto s iterative method to solve the oliear matrix equatio X + A X A = Q are that the fixed closed ball i which the uique solutio of the matrix equatio icluded ca be determied, the uique solutio i the fixed closed ball ca be obtaied, ad the expressio of the error estimate of the approximate solutio i the fixed closed ball ca be give. May tests show that, if the iitial matrix X 0 = Q, the sequece {X k } defied by (2.3) coverges to the maximal positive defiite solutio X L of the oliear matrixequatio(.). IftheiitialmatrixX 0 = (AQ A ) /, thesequece{x k } defiedby(2.3)covergestothemiimalpositivedefiite solutiox S oftheoliear matrix equatio (.). Here the maximal (miimal) solutio X L (X S ) meas that for everypositivedefiite solutioy ofthe oliearmatrix equatio(.) satisfiesx L Y (Y X S ). That is, X L Y (Y X S ) is a positive defiite matrix. Ufortuately these results ca ot be proved here. The disadvatage is that the rate of covergece is relatively slower tha some existig fixed poit iteratio methods. This is because the ier iteratio, that is, LSQR M algorithm may eed to iterate may times to achievethe required accuracy. Ackowledgemet. The authors thak the aoymous referee for valuable suggestios which helped them to improve this article.

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