ITERATIVE SOLUTION OF TWO MATRIX EQUATIONS

Transcription

1 ITERATIVE SOLUTION OF TWO MATRIX EQUATIONS CHUN-HUA GUO AND PETER LANCASTER Abstract We study iterative methods for fidig the maximal Hermitia positive defiite solutios of the matrix equatios X + A X 1 A = Q ad X A X 1 A = Q, where Q is Hermitia positive defiite Geeral covergece results are give for the basic fixed poit iteratio for both equatios Newto s method ad iversio free variats of the basic fixed poit iteratio are discussed i some detail for the first equatio Numerical results are reported to illustrate the covergece behaviour of various algorithms 1 Itroductio I this paper, we are cocered with the iterative solutio of the matrix equatios (11 X + A X 1 A = Q, ad (12 X A X 1 A = Q I both cases, the matrix Q is m m Hermitia positive defiite ad Hermitia positive defiite solutios are required These two equatios have bee studied recetly by several authors (see [1], [2], [3], [4], [16], [17] For the applicatio areas i which the equatios arise, see the refereces give i [1] ad [4] For Hermitia matrices X ad Y, we write X Y (X > Y if X Y is positive semidefiite (defiite A Hermitia solutio X + of a matrix equatio is called maximal if X + X for ay Hermitia solutio X of the matrix equatio A miimal solutio ca be defied similarly It is proved i [3] that if (11 has a positive defiite solutio, the it has a maximal Hermitia solutio X + ad a miimal Hermitia solutio X Ideed, we have 0 < X X X + for ay Hermitia solutio X of (11 Moreover, we have ρ(x+ 1 A 1 (see, eg, [16], where ρ( is the spectral radius Whe the matrix A is osigular, the miimal positive defiite solutio of (11 ca be foud via the maximal solutio of aother equatio of the same type (cf [3, Thm 33] I [3], a algorithm was preseted to fid the miimal solutio of the equatio (11 for the case where A is sigular The algorithm was based o a recursive reductio process The reductio process is useful i showig that the miimal positive defiite solutio of (11 exists eve if the matrix A is sigular However, it is usually impossible to fid the miimal solutio usig that algorithm The reaso is that the miimal solutio, as a fuctio of (A, Q, is geerally ot cotiuous at a sigular matrix A for fixed Q The situatio is already clear for 1991 Mathematics Subject Classificatio Primary 15A24, 65F10; Secodary 65H10, 93B40 Key words ad phrases matrix equatios, positive defiite solutio, fixed poit iteratio, Newto s method, covergece rate, matrix pecils 1

2 2 CHUN-HUA GUO AND PETER LANCASTER the scalar equatio X + ɛ 2 X 1 = 1 The miimal solutio of this equatio is ot cotiuous at ɛ = 0 We will therefore limit our discussio to the maximal solutio The equatio (12 is quite differet As the scalar case suggests, it always has a uique positive defiite solutio, which is the maximal solutio (see [4] I Sectio 2, we discuss the covergece behaviour of the basic fixed poit iteratio for the maximal solutio of (11 ad (12 I Sectio 3, we study the covergece behaviour of iversio free variats of the basic fixed poit iteratio I geeral, these algorithms are liearly coverget ad do ot perform well whe there are eigevalues of X+ 1 A o, or ear, the uit circle However, Newto s method ca be applied This has a global covergece property whe applied to (11 ad, although it is step-wise expesive, offers cosiderable advatages i this situatio Sectios 4 ad 5 are devoted to properties of the Newto iteratio I Sectio 6, we give matrix pecil descriptios for the eigevalues of X+ 1 A This admits the computatio of these eigevalues without prior kowledge of X + Some umerical examples are reported i Sectio 7 Throughout the paper, will be the spectral orm for square matrices uless otherwise oted 2 Basic fixed poit iteratio The maximal solutio X + of (11 ca be foud by the followig basic fixed poit iteratio: Algorithm 21 X 0 = Q, X +1 = Q A X 1 A, = 0, 1, For Algorithm 21, we have X 0 X 1, ad lim X = X + (see, eg, [3] The followig result is give i [16] Theorem 22 For ay ɛ > 0, X +1 X + ( X 1 + A + ɛ 2 X X + for all large eough We ow show that the above result ca be improved cosiderably Theorem 23 For all 0, Moreover, X +1 X + X 1 + A 2 X X + lim sup X X + (ρ(x+ 1 A 2 Proof Sice X +1 = Q A X 1 A ad X + = Q A X+ 1 A, we have Thus, X +1 X + =A (X 1 + X 1 A =A (X+ 1 + X 1 X+ 1 (X X + X+ 1 A =A X+ 1 (X X + X+ 1 A A X+ 1 (X X + X 1 (X X + X+ 1 A (21 0 X +1 X + A X 1 + (X X + X 1 + A

3 ITERATIVE SOLUTION OF TWO MATRIX EQUATIONS 3 Therefore, X +1 X + A X 1 + (X X + X 1 + A X 1 + A 2 X X + By repeated applicatio of (21, we get Hece, ad lim sup 0 X X + (A X 1 + (X 0 X + (X 1 + A X X + (X 1 + A 2 X 0 X +, X X + lim (X+ 1 A 2 X 0 X + = (ρ(x 1 + A 2 I the last equality, we have used the fact that lim B 1/ = ρ(b for ay square matrix B ad ay orm We metioed earlier that ρ(x+ 1 A 1 is always true From the secod part of the above result, we kow that the covergece of the fixed poit iteratio is R- liear wheever ρ(x+ 1 A < 1 For detailed defiitios of the rates of covergece, see [13] Zha asked i [16] whether ρ(x+ 1 A 1 implies X+ 1 A 1 This is ot the case ad, i fact, it is possible to have X+ 1 A > 1 whe ρ(x+ 1 A < 1 If ρ(x+ 1 A = 1, the covergece of the fixed poit iteratio is typically subliear Example 21 We cosider the scalar case of (11 with A = 1 2 ad Q = 1, ie, X X 1 = 1 Clearly, X + = 1 2 ad ρ(x 1 + A = 1 For the fixed poit iteratio X 0 = 1, X +1 = X 1, = 0, 1,, we have X 0 > X 1 >, ad lim X = 1 2 Note that Thus, X = X 1 2 (X X 1 lim ie, the covergece is subliear X 1 2 = lim X X 1 2 = 1, For the matrix equatio (12, the maximal solutio X + ca also be foud by a fixed poit iteratio similar to that of Algorithm 21 Thus, we cosider: Algorithm 24 X 0 = Q, X +1 = Q + A X 1 A, = 0, 1, For Algorithm 24, we have X 0 X 2 X 4, X 1 X 3 X 5, ad lim X = X + (see [4] The followig result is immediate (cf Theorem 22

4 4 CHUN-HUA GUO AND PETER LANCASTER Theorem 25 For Algorithm 24 ad ay ɛ > 0, X +1 X + ( X 1 + A 2 + ɛ X X + for all large eough However, more ca be said about Algorithm 24 Theorem 26 For Algorithm 24, we have (22 X 2k X + X 1 + A 2 X 2k 1 X + for all k 1, ad Proof Observe that We have lim sup X X + (ρ(x+ 1 A 2 < 1 X +1 X + = A X 1 (X X + X+ 1 A = A X+ 1 (X X + X+ 1 A + A X+ 1 (X X + X 1 (X X + X+ 1 A (23 0 X + X 2k A X 1 + (X 2k 1 X + X 1 + A, from which (22 follows We also have X 2k+1 X + =A X 1 + (X + X 2k X 1 + A + A X 1 + (X + X 2k X 1 2k (X + X 2k X 1 + A =A X 1 + (X + X 2k 1/2 [I + (X + X 2k 1/2 X 1 2k (X + X 2k 1/2 ](X + X 2k 1/2 X 1 + A For ay ɛ > 0, there exists a k 0 such that (X + X 2k 1/2 X 1 2k (X + X 2k 1/2 ɛi for all k k 0 Therefore, (24 0 X 2k+1 X + (1 + ɛa X 1 + (X + X 2k X 1 + A for all k k 0 Combiig (23 ad (24, we obtai for k k 0 0 X 2k+1 X + (1 + ɛ k k0+1 (A X 1 + 2(k k0+1 (X + X 2k0 (X 1 + A 2(k k0+1 Thus, Similarly, X 2k+1 X + (1 + ɛ k k0+1 (X 1 + A 2(k k0+1 2 X + X 2k0 X 2k+2 X + (1 + ɛ k k0+1 (X 1 + A 2(k k0+2 2 X + X 2k0 Therefore, lim sup X X ɛ (ρ(x+ 1 A 2 Sice ɛ > 0 is arbitrary, we have Sice lim sup X X + (ρ(x+ 1 A 2 X + (X 1 + A X + (X 1 + A = Q,

5 ITERATIVE SOLUTION OF TWO MATRIX EQUATIONS 5 with Q > 0 ad X + > 0, we have ρ(x 1 + A < 1 (see, eg, [12, p 451] 3 Iversio free variat of the basic fixed poit iteratio I [16], Zha proposed a iversio free variat of the basic fixed poit iteratio for the maximal solutio of (11 whe Q = I For geeral positive defiite Q, Zha s algorithm takes the followig form: Algorithm 31 Take X 0 = Q, Y 0 = Q 1 For = 0, 1,, compute X +1 = Q A Y A, Y +1 = Y (2I X Y The covergece of Algorithm 31 was established i [16] for Q = I result ca easily be trasplated ad we have Zha s Theorem 32 If (11 has a positive defiite solutio the, for Algorithm 31, X 0 X 1, Y 0 Y 1, ad lim X = X +, lim Y = X 1 + The problem of covergece rate for Algorithm 31 was ot solved i [16] We ow establish the followig result: Theorem 33 For ay ɛ > 0, we have (31 Y +1 X 1 + ( AX ɛ 2 Y 1 X 1 + ad (32 X +1 X + A 2 Y X 1 + for all large eough If A is osigular, we also have (33 X +1 X + ( X 1 + A + ɛ 2 X 1 X + for all large eough Proof We have from Algorithm 31 Thus, (34 Y +1 = Y (2I (Q A Y 1 AY = 2Y Y QY + Y A (X Y 1 X 1 + AY = 2Y Y X + Y + Y A (Y 1 X 1 + AY X 1 + Y +1 = X 1 + Y + Y X + Y Y + Y A (X 1 + Y 1 AY = (X 1 + Y X + (X 1 + Y + Y A (X 1 + Y 1 AY The iequality (31 follows sice Y X 1 + Y 1 X 1 + ad lim Y = X 1 + The iequality (32 is true sice (35 X +1 X + = A (X 1 + Y A If A is osigular, we have by (34 ad (35 X +1 X + = (X X + A 1 X + (X 1 + Y 1 A + A Y 1 (X 1 X + Y 1 A Therefore, sice X X + X 1 X +, (33 is true for all large eough The above proof shows that Algorithm 31 should be modified as follows to improve the covergece properties:

6 6 CHUN-HUA GUO AND PETER LANCASTER Algorithm 34 Take X 0 = Q, 0 < Y 0 Q 1 For = 0, 1,, compute Y +1 = Y (2I X Y, X +1 = Q A Y +1 A Note that oe coveiet choice of Y 0 is Y 0 = I/ Q We ca also use this choice of Y 0 i Algorithm 31 Theorems 32 ad 33 remai true for ay Y 0 such that 0 < Y 0 Q 1 Lemma 35 (cf [16] If C ad P are Hermitia matrices of the same order with P > 0, the CP C + P 1 2C Theorem 36 If (11 has a positive defiite solutio ad {X }, {Y } are determied by Algorithm 34, the X 0 X 1, lim X = X + ; Y 0 Y 1, lim Y = X 1 + Proof It is clear that (36 X 0 X 1 X X +, Y 0 Y 1 Y X 1 + is true for = 1 Assume (36 is true for = k We have by Lemma 35 Therefore, Sice Y k X 1 Y k+1 = 2Y k Y k X k Y k X 1 k X 1 + X k+1 = Q A Y k+1 A Q A X+ 1 A = X + 1, we have Y X k Thus, k 1 X 1 k k Y k+1 Y k = Y k (Y 1 k X k Y k 0, ad X k+1 X k = A (Y k+1 Y k A 0 We have ow proved (36 for = k + 1 Therefore, (36 is true for all, ad the limits lim X ad lim Y exist As i [16], we have lim X = X +, ad lim Y = X+ 1 Theorem 37 For Algorithm 34 ad ay ɛ > 0, we have ad Y +1 X 1 + ( AX ɛ 2 Y X 1 + (37 X X + A 2 Y X 1 + for all large eough If A is osigular, we also have for all large eough X +1 X + ( X 1 + A + ɛ 2 X X + Proof The proof is very similar to that of Theorem 33 We see from the estimates i Theorem 33 ad Theorem 37 that Algorithm 34 ca be faster tha Algorithm 31 by a factor of 2 Compared with Algorithm 21, Algorithm 34 eeds more computatioal work per iteratio However, Algorithm 34 has better umerical properties sice matrix iversios have bee avoided Algorithm 34 is particularly useful o a parallel computig system, sice matrix-matrix multiplicatio ca be carried out i parallel very efficietly (see, eg, [6] For Algorithm 34, R-liear covergece ca be guarateed wheever ρ(x+ 1 A < 1 This will be a cosequece of the followig geeral result

7 ITERATIVE SOLUTION OF TWO MATRIX EQUATIONS 7 Theorem 38 (cf [10, p 21] Let T be a (oliear operator from a Baach space E ito itself ad x E be a solutio of x = T x If T is Fréchet differetiable at x with ρ(t x < 1, the the iterates x +1 = T x ( = 0, 1, coverge to x, provided that x 0 is sufficietly close to x Moreover, for ay ɛ > 0, x x c(x 0 ; ɛ(ρ(t x + ɛ, where is the orm i E ad c(x 0 ; ɛ is a costat idepedet of Corollary 39 For Algorithm 34, we have (38 lim sup X X + (ρ(x+ 1 A 2 Proof For Algorithm 34, we have Y +1 = T (Y, = 0, 1,, where the operator T is defied o C m m (m is the order of Q by T (Y = 2Y Y QY + Y A Y AY It is foud that the Fréchet derivative T Y : Cm m C m m is give by Therefore, T Y (Z = 2Z ZQY Y QZ + ZA Y AY + Y A Y AZ + Y A ZAY The spectrum of T X 1 + we have T (Z = X X A ZAX+ 1 + cosists of eigevalues oly If λ is a eigevalue of T, X 1 + (39 X 1 + A ZAX 1 + = λz for some Z 0 If λ > (ρ(x+ 1 A 2, the equatio (39 would have zero as the oly solutio (see, eg [11, p 100] Therefore, ρ(t (ρ(x X A 2 I fact, we have + ρ(t = (ρ(x X A 2, sice (39 has a ozero solutio for λ = (ρ(x+ 1 A 2 + By Theorem 38, we have lim sup I view of (37, we also have (38 Y X+ 1 ρ(t X 1 + = (ρ(x 1 + A 2 For equatio (12, we ca also have a algorithm similar to Algorithm 34 However, the algorithm is ot always coverget 4 Prelimiaries o Newto s method For equatios (11 ad (12, the covergece of the algorithms i the above two sectios may be very slow whe X+ 1 A has eigevalues close to (or eve o the uit circle I these situatios, Newto s method ca be recommeded The equatio (11 is a special discrete algebraic Riccati equatio (DARE, if we are willig to relax certai restrictios ormally imposed o such equatios for the purpose of aalysis Therefore, we will start with a review of some previous results o Newto s method for DAREs We cosider a DARE of the form (41 X + A XA + Q (C + B XA (R + B XB 1 (C + B XA = 0,

8 8 CHUN-HUA GUO AND PETER LANCASTER where A, Q C, B C m, C C m, R C m m, ad Q = Q, R = R We deote by R(X the left-had side of (41 Let H be the set of Hermitia matrices i C ad let D = {X H R + B XB is ivertible} We have R : D H It is assumed throughout that D is oempty ad that there is a X D such that R + B XB > 0 The first Fréchet derivative of R at a matrix X D is a liear map R X : H H give by (42 R X(H = H +  HÂ, where  = A B(R + B XB 1 (C + B XA For A C ad B C m, the pair (A, B is said to be d-stabilizable if there is a K C m such that A BK is d-stable, ie, all its eigevalues are i the ope uit disk The followig result is a modificatio of Theorem 1311 i [11] It has bee oted i [7] that the matrix R does ot eed to be ivertible Theorem 41 Let (A, B be a d-stabilizable pair ad assume that there is a Hermitia solutio X of the iequality R(X 0 for which R+B XB > 0 The there exists a maximal Hermitia solutio X + of R(X = 0 Moreover, R + B X + B > 0 ad all the eigevalues of A B(R + B X + B 1 (C + B X + A lie i the closed uit disk The Newto method for the solutio of (41 is (43 X i = X i 1 (R X i 1 1 R(X i 1, i = 1, 2,, give that the maps R X i (i = 0, 1, are all ivertible Whe we apply Newto s method to the DARE (41 with (A, B d-stabilizable, the iitial matrix X 0 is chose so that A B(R + B X 0 B 1 (C + B X 0 A is d- stable The usual way to geerate such a X 0 is as follows We choose L 0 C m such that A 0 = A BL 0 is d-stable, ad the take X 0 to be the uique solutio of the Stei equatio (44 X 0 A 0X 0 A 0 = Q + L 0RL 0 C L 0 L 0C I view of (42, the Newto iteratio (43 ca be rewritte as (45 X i A i X i A i = Q + L i RL i C L i L i C, i = 1, 2,, where ad L i = (R + B X i 1 B 1 (C + B X i 1 A A i = A BL i Theorem 42 Uder the coditios of Theorem 41 ad for ay L 0 C m such that A 0 = A BL 0 is d-stable, startig with the Hermitia matrix X 0 determied by (44, the recursio (45 determies a sequece of Hermitia matrices {X i } i=0 for which A B(R+B X i B 1 (C+B X i A is d-stable for i = 0, 1,, X 0 X 1, ad lim i X i = X + A importat feature of Newto s method applied to the Riccati equatio is that the covergece is ot local The applicatio of Newto s method to the Riccati equatio was iitiated i [8] uder some coditios which, with the wisdom of hid-sight, are see to be restrictive Similarly, Theorem 42 was established i the proof of [14, Thm 31] uder the additioal coditio that R > 0 The positive defiiteess of R was replaced by the ivertibility of R i the proof of [11, Thm 1311] It has bee oted i [7] that the ivertibility of R is also uecessary It is

9 ITERATIVE SOLUTION OF TWO MATRIX EQUATIONS 9 the removal of this restrictio that will allow its applicatio to the matrix equatio (11 Theorem 43 (cf [7] If A B(R + B X + B 1 (C + B X + A is d-stable i Theorem 42, the the sequece {X i } i=0 coverges to X + quadratically Theorem 44 (cf [7] Uder the coditios i Theorem 41 ad assumig that all eigevalues of A B(R+B X + B 1 (C+B X + A o the uit circle are semisimple, the Newto sequece {X i } coverges to X + either quadratically, or liearly with rate 1/2 5 Applicatios of Newto s method We ow let m = i DARE (41, ad take A = 0, R = 0, B = I The equatio becomes X + C X 1 C = Q, which has the same form as (11, ad the hypotheses of Theorem 41 are trivially satisfied wheever it has a positive defiite solutio We ca the apply the results we have just reviewed to the equatio (11 (the matrix A i (11 has take the place of the matrix C i (41 The ext result is a immediate cosequece of Theorem 41 The first coclusio has bee proved i [3] The secod coclusio has bee oted i [16] Theorem 51 If (11 has a positive defiite solutio, the it has a maximal positive defiite solutio X + ad ρ(x 1 + A 1 By takig L 0 = 0 i (44, we obtai A 0 = 0 (which is certaily d-stable ad the followig algorithm for equatio (11: Algorithm 52 (Newto s method for (11 Take X 0 = Q For i = 1, 2,, compute L i = X 1 i 1A, ad solve (51 X i L i X i L i = Q 2L i A Note that the Stei equatio (51 is uiquely solvable whe ρ(l i < 1 From Theorems 42, 43 ad 44 we have: Theorem 53 If (11 has a positive defiite solutio, the Algorithm 52 determies a sequece of Hermitia matrices {X i } i=0 for which ρ(l i < 1 for i = 0, 1,, X 0 X 1, ad lim i X i = X + The covergece is quadratic if ρ(x+ 1 A < 1 If ρ(x+ 1 A = 1 ad all eigevalues of X+ 1 A o the uit circle are semisimple, the the covergece is either quadratic or liear with rate 1/2 Note also that if there are eigevalues of X+ 1 A o the uit circle ad liear covergece is idetified the, as show i [7], a double-step modificatio of Newto s method ca be used to great advatage We ow tur our attetio to the equatio (12 I [4] it is show that, if A is osigular, the maximal solutio X + of (12 is also the maximal Hermitia solutio of the DARE X = Q + F XF F X(R + X 1 XF, where F = A A 1, R = AQ 1 A The maximal solutio X + ca the be foud by direct applicatio of Newto s method for the DARE (41 However, the overhead costs of Newto s method are higher tha for equatio (11, ad compariso with the basic fixed poit iteratio is less favorable If we apply Newto s method directly to equatio (12, covergece caot be guarateed if the iitial guess is ot close to X + However, Newto s method still has local quadratic covergece ad ca be used as a efficiet correctio method

10 10 CHUN-HUA GUO AND PETER LANCASTER Algorithm 54 (Newto correctio for (12 For give X k sufficietly close to X + ad i = k + 1, k + 2,, compute L i = X 1 i 1A, ad solve (52 X i + L i X i L i = Q + 2L i A The equatios (51 ad (52 ca be solved by a complex versio of the algorithm described i [5] The computatioal work per iteratio for Algorithm 52 or 54 is roughly times that for Algorithm 21 or 24 I cotrast to the equatio (51, the equatio (52 is ot ecessarily early sigular whe X+ 1 A has eigevalues very close to the uit circle This makes the Newto correctio eve more attractive 6 Matrix pecils As we have see i the previous sectios, the covergece rates of various algorithms for equatio (11 or (12 are depedet o the eigevalues of X+ 1 A, where X + is the solutio of (11 or (12 that we seek I this sectio, we will relate the eigevalues of X+ 1 A to the eigevalues of a matrix pecil which is idepedet of X + As we have see, the equatio (11 is a special case of the DARE (41 For (41 we cosider the matrix pecil λf e G e with I 0 0 A 0 B F e = 0 A 0, G e = Q I C 0 B 0 C 0 R Matrix pecils of this type were first itroduced i [15] Theorem 61 (cf [7] If (41 has a Hermitia solutio X, the λf e G e is a regular pecil Moreover, α is a eigevalue of (61 A B(R + B XB 1 (C + B XA if ad oly if α ad ᾱ 1 are eigevalues of λf e G e If we assume further that (A, B is d-stabilizable ad R + B XB > 0, the a uimodular α is a eigevalue of (61 with partial multiplicity k if ad oly if it is a eigevalue of λf e G e with partial multiplicity 2k Corollary 62 For equatio (11, the eigevalues of X+ 1 A are precisely the eigevalues of the matrix pecil λf 1 G 1 λ I I I Q I A A 0 0 iside or o the uit circle, with half of the partial multiplicities for each eigevalue o the uit circle Accordig to [3, Thm 21], the equatio (11 has a positive defiite solutio if ad oly if the ratioal matrix-valued fuctio ψ(λ = Q + λa + λ 1 A is regular (ie, detψ(λ 0 for some λ ad ψ(λ 0 for all λ o the uit circle I particular, (11 has a positive defiite solutio if ψ(λ > 0 for all λ o the uit circle Let r(t be the umerical radius of T C m m, defied by r(t = max{ x T x : x C m, x x = 1} Note that r(t T 2r(T (see [9], for example The followig lemma has bee proved i [3]

11 ITERATIVE SOLUTION OF TWO MATRIX EQUATIONS 11 Lemma 63 ψ(λ > 0 for all λ o the uit circle if ad oly if r(q 1/2 AQ 1/2 < 1 2 As we have see i Theorem 53, the covergece of Algorithm 52 is quadratic if ρ(x 1 + A < 1 Our fial theorem clarifies this coditio Theorem 64 For equatio (11, ρ(x 1 + A < 1 if ad oly if r(q 1/2 AQ 1/2 < 1 2 Proof By Corollary 62, it is eough to show r(q 1/2 AQ 1/2 < 1 2 if ad oly if the pecil λf 1 G 1 has o eigevalues o the uit circle By appropriate block elimiatio we fid that det(λf 1 G 1 = det λi 0 I Q I A A λi I = det Q λa I A A λi 0 ( Q λa I = det A λi ( Q λa = det I A + λ(q + λa 0 = ( 1 m λ m det(q + λ 1 A + λa Therefore, λf 1 G 1 has o eigevalues o the uit circle if ad oly if ψ(λ > 0 for all λ o the uit circle, the latter is equivalet to r(q 1/2 AQ 1/2 < 1 2 by Lemma 63 We ow tur our attetio to the equatio (12 I this case, the eigevalues of + A are related to the eigevalues of the matrix pecil λf 2 G 2 with F 2 = I , G 2 = 0 0 I Q I A 0 I 0 A 0 0 X 1 Lemma 65 If X is a solutio of (12, the I 0 0 (λf 2 G 2 X I 0 = I 0 0 I 0 A X 1 X 1 A 0 I X 0 I where M = I A X I 0, N = X 1 A 0 I 0 I X (λm N, Proof The result is easily verified by direct computatio Corollary 66 The eigevalues of X 1 + A are precisely the eigevalues of λf 2 G 2 iside the uit circle

12 12 CHUN-HUA GUO AND PETER LANCASTER Proof Takig X = X + i the above Lemma, we get det(λf 2 G 2 = det(λm N = det(x + det(λi X 1 + Adet(λA X I Sice ρ(x+ 1 A < 1, the zeros of det(λa X+ 1 + I = 0 are outside the uit circle The coclusio i the corollary follows readily 7 Numerical results I this sectio, we give some examples to illustrate the covergece behaviour of various algorithms we have discussed Double precisio is used i all computatios Example 71 Cosider equatio (11 with ( ( A =, Q = The maximal solutio (with the first 9 digits is foud to be ( X + = We compare the umber of iteratios required for Algorithms 21, 31 ad 34 to get the first 6 correct digits Algorithm 21 eeds 16 iteratios with X 16 = ( Algorithm 31 eeds 34 iteratios with ( X 34 = Algorithm 34 eeds 19 iteratios with ( X 19 = We have used Y 0 = I/ Q for Algorithms 31 ad 34 If we use Y 0 = Q 1, the umbers of iteratios are 32 ad 17, respectively The covergece is liear for all three algorithms The covergece of Algorithm 21 is slightly faster tha that of Algorithm 34, while the covergece of Algorithm 34 is faster tha that of Algorithm 31 by roughly a factor of 2 These are cosistet with the covergece results i Sectio 2 ad Sectio 3 For this example, we have ρ(x+ 1 A = 06708, X+ 1 A = 13829, ad AX+ 1 = The ext two examples will show that, for equatio (11, Algorithm 52 ca be much more efficiet tha Algorithm 21 Of course, for easy problems the basic fixed poit iteratio eeds o more tha 30 iteratios to get a good approximate solutio I these cases we caot expect Newto s method to perform better, sice two or three iteratios are usually ecessary for the Newto iteratio For these two examples, we use the practical stoppig criterio (71 X + A X 1 A Q < ɛ for both Algorithm 52 ad Algorithm 21, where ɛ is a prescribed tolerace

13 ITERATIVE SOLUTION OF TWO MATRIX EQUATIONS 13 Example 72 We cosider the equatio (11 with Q = I ad A = For this example, A is Hermitia (ad hece ormal The exact maximal solutio ca be foud accordig to the formula X + = 1 2 [I + (I 4A A 1/2 ], which is valid for ay ormal matrix A with A 1/2 (see [17] Sice r(a = A = 1/2 for this example, we have ρ(x+ 1 A = 1 (cf Thm 64 The covergece of Algorithm 21 turs out to be subliear It eeds 7071 iteratios to satisfy (71 for ɛ = 10 8, with X7071 F = symm O the other had, the covergece of Algorithm 52 is liear with rate 1/2 (cf Thm 53 The stoppig criterio is satisfied after 12 iteratios, with X12 N = symm We fid that both X7071 F ad X12 N have four correct digits, with X12 N slightly better If we use a double Newto step followig X12, N the resultig approximate solutio with the first 8 digits (without roudig is X 13 = symm All digits are the same as i the exact solutio This example shows that Newto s method ca be much more efficiet tha the basic fixed poit iteratio whe r(q 1/2 AQ 1/2 is equal or very close to 1/2 Example 73 We cosider the equatio (11 with A = , Q = For this example, r(q 1/2 AQ 1/2 < 1/2 Thus the Newto iteratio coverges quadratically to the maximal solutio It eeds 8 iteratios to satisfy the stoppig criterio (71 for ɛ = The computed maximal solutio is X + = symm The basic fixed poit iteratio eeds 332 iteratios to satisfy the same criterio The covergece is liear sice ρ(x+ 1 A < 1 Note that X+ 1 A > 1 for this example The last example is devoted to equatio (12

14 14 CHUN-HUA GUO AND PETER LANCASTER Example 74 Cosider equatio (12 with ( A = 10 60, Q = ( The maximal solutio ca be foud to be ( X + = Usig Algorithm 24, we get after 100 iteratios ( X 100 = After 300 more iteratios, we get ( X 400 = If we use Newto correctio (Algorithm 54 startig with X 100, we get after two iteratios ( X 102 =, which is already slightly better tha X 400 give above The slow covergece of Algorithm 24 is cosistet with the secod coclusio i Theorem 26, sice we have ρ(x+ 1 A = for this example For this example, we have X+ 1 A > 1 Therefore, Theorem 25 ad the first coclusio i Theorem 26 are useless Refereces 1 W N Aderso, Jr, T D Morley, ad G E Trapp, Positive solutios to X = A BX 1 B, Liear Algebra Appl 134 (1990, J C Egwerda, O the existece of a positive defiite solutio of the matrix equatio X + A T X 1 A = I, Liear Algebra Appl 194 (1993, J C Egwerda, A C M Ra, ad A L Rijkeboer, Necessary ad sufficiet coditios for the existece of a positive defiite solutio of the matrix equatio X + A X 1 A = Q, Liear Algebra Appl 186 (1993, A Ferrate ad B C Levy, Hermitia solutios of the equatio X = Q + NX 1 N, Liear Algebra Appl 247 (1996, J D Gardier, A J Laub, J J Amato, ad C B Moler, Solutio of the Sylvester matrix equatio AXB T + CXD T = E, ACM Tras Math Software 18 (1992, G H Golub ad C F Va Loa, Matrix computatios, Third editio, Johs Hopkis Uiversity Press, Baltimore, MD, C-H Guo, Newto s method for discrete algebraic Riccati equatios whe the closed-loop matrix has eigevalues o the uit circle, SIAM J Matrix Aal Appl, to appear 8 G A Hewer, A iterative techique for the computatio of the steady-state gais for the discrete optimal regulator, IEEE Tras Autom Cotrol 16 (1971, R A Hor ad C R Johso, Topics i matrix aalysis, Cambridge Uiversity Press, Cambridge, M A Krasoselskii, G M Vaiikko, P P Zabreiko, Ya B Rutitskii, ad V Ya Stetseko, Approximate solutio of operator equatios, Wolters-Noordhoff Publishig, Groige, P Lacaster ad L Rodma, Algebraic Riccati equatios, Claredo Press, Oxford, P Lacaster ad M Tismeetsky, The theory of matrices, Secod editio, Academic Press, Orlado, FL, J M Ortega ad W C Rheiboldt, Iterative solutio of oliear equatios i several variables, Academic Press, New York, A C M Ra ad R Vreugdehil, Existece ad compariso theorems for algebraic Riccati equatios for cotiuous- ad discrete-time systems, Liear Algebra Appl 99 (1988, 63 83

15 ITERATIVE SOLUTION OF TWO MATRIX EQUATIONS P Va Doore, A geeralized eigevalue approach for solvig Riccati equatios, SIAM J Sci Comput 2 (1981, X Zha, Computig the extremal positive defiite solutios of a matrix equatio, SIAM J Sci Comput 17 (1996, X Zha ad J Xie, O the matrix equatio X + A T X 1 A = I, Liear Algebra Appl 247 (1996, Departmet of Mathematics ad Statistics, Uiversity of Calgary, Calgary, Alberta, Caada T2N 1N4 Curret address: Departmet of Computer Sciece, Uiversity of Califoria, Davis, Califoria address: guo@csucdavisedu Departmet of Mathematics ad Statistics, Uiversity of Calgary, Calgary, Alberta, Caada T2N 1N4 address: lacaste@ucalgaryca