Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations

Transcription

1 Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations Wen-Wei Lin Shu-Fang u Abstract In this paper a structure-preserving transformation of a symplectic pencil is introduced referred to as the doubling transformation and its some basic properties are presented. Based on the nice properties of this kind of transformations a unified convergence theory for the structure-preserving doubling algorithms for solving a class of Riccati-type matrix equations is established by using only the knowledge from elementary matrix theory. Key words. matrix equation structure-preserving doubling algorithm convergence rate AMS subject classfications. 15A24 65H10 93B50 93D15 1 Introduction In this paper we investigate the convergence of the structure-preserving doubling algorithms (SDA) for the computation of the symmetric positive definite or semi-definite solutions to the following Riccati-type matrix equations: Continuous-time algebraic Riccati equation (CARE) G + A T + A + H = 0 (1.1) where AHG R n n with G and H symmetric positive semi-definite. Discrete-time algebraic Riccati equation (DARE) = A T (I + G) 1 A + H (1.2) where AHG R n n with G and H symmetric positive semi-definite. Nonlinear matrix equation with the plus sign (NME-P) 3 where AQ R n n with Q symmetric positive definite. Nonlinear matrix equation with the minus sign (NME-M) 11 where AQ R n n with Q symmetric positive definite. + A T 1 A = Q (1.3) A T 1 A = Q (1.4) Department of Mathematics National Tsing Hua University Hsinchu 300 Taiwan (wwlin@am.nthu.edu.tw). School of Mathematical Sciences Peking University Beijing China (xsf@pku.edu.cn). This research was supported in part by the National Center for Theoretical Sciences in Taiwan. 1

2 The Riccati-type matrix equations occur in many important applications (see and references therein). The nonlinear matrix equations CARE and DARE have been studied extensively (see and 31); and the nonlinear matrix equations NME-P and NME-M has been studied recently by several authors (see and 32). A class of methods referred to as the doubling algorithms has attracted much interests in 70s and 80s (see 2 and references therein). These methods originate from the fixed-point iteration derived from the DARE: k+1 = A T k (I + G k ) 1 A + H. Instead of generating the sequence { k } doubling algorithms generate { 2 k}. Doubling algorithms were largely forgotten in the past decade. Recently they have been revived for DAREs and CAREs because their nice numerical behavior: quadratical convergence rate low cost computational cost per step and good numerical stability (see 6 7 8). Concerning the matrix equations NME-Ps and NME-Ms in 2001 B. Meini proposed an iterative method with the same numerical behavior as the SDA algorithms for solving the DAREs and CAREs (see 26 16). In this paper by employing the same technique as in 8 we derive two SDA algorithms for solving the NME-Ps and NME-Ms as a result we find they are essentially the same as Meini s in 26. In summary we find all the SDA algorithms for solving the four Riccati-type matrix equations can be viewed as applying some special structure-preserving transformations to the associated symplectic pencils repeatedly. Therefore we first introduce a structure-preserving transformation of a symplectic pencil which is referred to as the doubling transformation and then prove that an important feature of this kind of transformations is that it is structure preserving eigenspace preserving and eigenvalue doubling. Finally based on the nice properties of the doubling transformations we develop a unified convergence theory for these SDA algorithms by using only the knowledge from elementary matrix theory. Throughout this pater the symbols 2 denote the matrix spectral norm. For a given n n matrix A we use ρ(a) to denote the spectral radius of A. For real symmetric matrices and Y we write > Y ( Y ) if Y is symmetric positive definite (semi-definite). The paper is organized as follows. In Section 2 we introduce a structure-preserving transformation of a symplectic pencil and develop its basic properties. In Section 3 we do the convergence analysis of the SDA algorithm for solving the DAREs based on the theory established in Section 2. Section 4 derives a SDA algorithm for solving the NME-P (1.3) by using the doubling transformations and establishes the convergence theory of the alorithm. Concluding remarks are given in Section 5. 2 Doubling Transformation In this section we introduce a structure-preserving transformation of a symplectic pencil and develop its some basic properties. We begin with the definition of the transformation. Let M λl R 2n 2n be a symplectic pencil i.e. MJM T = LJL T J = 0 I I 0. (2.1) Define { N(ML) = M L : M L R 2n 2n L } rankm L = 2n M L = 0. (2.2) M L Since rank 2n it follows that N(ML). For any given M M L N(ML) define M = M M L = L L. (2.3) The transformation M λl M λ L 2

3 is called a doubling transformation. An important feature of this kind of transformations is that it is structure preserving eigenspace preserving and eigenvalue doubling as shown in the following theorem. Theorem 2.1. Assume that the pencil M λ L is a doubling transformation of a symplectic pencil M λl. Then we have: (a) The pencil M λ L is still symplectic. U U (b) If M = L S where UV R V V n m and S R m m then M (c) If the pencil M λl has the Kronecker canonical form Jr 0 Ir 0 WMZ = WLZ = 0 I 2n r 0 N 2n r U = V L U S V 2. (2.4) where WZ are nonsingular J r a Jordan matrix corresponding to the finite eigenvalues of M λl and N 2n r a nilpotent Jordan matrix corresponding to the infinite eigenvalues of M λl then there exists a nonsingular matrix Ŵ such that Ŵ MZ J 2 = r 0 Ŵ 0 I LZ Ir 0 = 2n r 0 N2n r 2. (2.5) Proof. (a) Since M L N(ML) implies that M L = L M it follows from (2.1) that MJ M T = M MJM T M T = M LJL T M T = (L M)J(M T L T ) = L LJL T L T = LJ L T which means that the pencil M λ L is symplectic. (b) Using the equality M L = L M again we have U U M L S = L V M S V and hence (c) Let M U = M V M U = M V L U S = L V M U S = L V L Jr 0 Ir 0 ˇM = W Ľ 0 I = W. 2n r 0 N 2n r U S 2 = V L U S 2. V Then we have Jr 0 Jr 0 Ir 0 Jr 0 ˇM LZ = WLZ = = 0 I 2n r 0 I 2n r 0 N 2n r 0 N 2n r Ir 0 Ir 0 Jr 0 Jr 0 Ľ MZ = WMZ = =. 0 N 2n r 0 N 2n r 0 I 2n r 0 N 2n r This together with Z nonsingular implies that ˇM L Ľ = 0. It is clear that rank M ˇM Ľ = M 2n. Notice that (2.4) implies that the pencil M λl is regular it must have rank = 2n. L Thus the rows of M L and ˇM M Ľ form two different bases of the left null space of. L Consequently there exists a nonsingular matrix Ŵ such that ŴM L = ˇM Ľ. It can be easily verified that the matrix Ŵ satisfies (2.5). 3

4 Remark 2.1. A subspace W of R 2n is called a generalized eigenspace of a pencil M λl R 2n 2n U if W is spanned by the columns of W = where UV R V n m W has full column rank and satisfies MW = LWS with S R m m. Therefore (b) of Theorem 2.1 tells us that if W is a generalized eigenspace of a symplectic pencil M λl then it is also a generalized eigenspace of its doubling transformation which is a corner stone for us to establish convergence theory of the SDA algorithms for solving the Riccati-type matrix equations in the next two sections. Remark 2.2. A pencil M λl is called regular if det(m λl) does not vanish identically. It is well-known that a pencil is regular if and only if it has a Kronecker canonical form. Thus (c) of Theorem 2.1 implies that if M λl is regular then its doubling transformation is also regular and that λ is a eigenvalue of M λl if and only if λ 2 is an eigenvalue of its doubling transformation. A symplectic pencil M λl is said to be a first standard symplectic form (SSF-1) if it has the following form A 0 I G M = L = H I 0 A T (2.6) with HG 0; A symplectic pencil M λl is said to be a second standard symplectic form (SSF-2) if it has the following form A 0 P I M = L = Q I A T (2.7) 0 with PQ 0. The following theorem shows that the two standard symplectic forms are preserved by an appropriate choice of the doubling transformations. Theorem 2.2. (a) Let M λl be a SSF-1 form. Then a matrix M L N(ML) can be constructed such that its corresponding doubling transformation M λ L is still a SSF-1. (b) Let M λl be a SSF-2 form. If Q P > 0 and Q A T (Q P) 1 A 0 then a matrix M L N(ML) can be constructed such that its corresponding doubling transformation M λ L is still a SSF-2. L Proof. (a) Applying block Gaussian elimination and row permutation to the matrix we M can get such that M = A(I + GH) 1 0 A T (I + HG) 1 H I I AG(I + HG) 1 L = 0 A T (I + HG) 1 (2.8) M L = L M (2.9) i.e. M L N(ML). Here the Sherman-Morrison-Woodbury formula (see e.g. 12 p. 50) is used. For more details see 8. We then compute L L and M M to produce  0 I Ĝ M = M M = Ĥ I L = L L = 0  T (2.10) where  = A(I + GH) 1 A (2.11) Ĝ = G + AG(I + HG) 1 A T (2.12) Ĥ = H + A T (I + HG) 1 HA. (2.13) It is clear that the resulting pencil is still a SSF-1. (b) Similarly under the condition Q P > 0 we can compute a matrix M L N(ML) with A(Q P) M = 1 0 I A(Q P) 1 A T (Q P) 1 L I = 0 A T (Q P) 1. (2.14) 4

5 Direct calculation gives rise to where  0 M = M M = Q I L = L L = P I  T 0 (2.15)  = A(Q P) 1 A (2.16) Q = Q A T (Q P) 1 A (2.17) P = P + A(Q P) 1 A T. (2.18) The assumption Q A T (Q P) 1 A 0 implies that the resulting pencil is still a SSF-2. Remark 2.3. The proof of Theorem 2.2 provided us with the well defined computation formulae for constructing the special structure preserving doubling transformations which is the base for us to derive the SDA algorithms for solving the Riccati-type matrix equations. 3 SDA Algorithms for Preserving the SSF-1 In this section we shall use the theory established in the last section to develop the convergence theory of the SDA algorithms for for solving the DAREs and CAREs. The SDA algorithms proposed in 8 and 7 will present in the next two subsections by using the doubling transformations defined in the last section. 3.1 SDA Algorithm for Solving the DAREs It is easy to verify that the DARE (1.2) has a symmetric positive semi-definite solution (i.e. 0) if and only if satisfies that I I M = L S (3.1) for some matrix S R n n where M = A 0 H I L = I G 0 A T. (3.2) Notice that the pencil M λl is just a SSF-1 form. Therefore applying the special doubling transformation defined by (2.11) (2.13) repeatedly gives rise to the following structure-preserving doubling algorithm. Algorithm SDA-1. A 0 = A G 0 = G H 0 = H A k+1 = A k (I + G k H k ) 1 A k G k+1 = G k + A k G k (I + H k G k ) 1 A T k H k+1 = H k + A T k (I + H k G k ) 1 H k A k. This is the SDA algorithm described in 8 in which extensive numerical experiments show that this algorithm is efficient and competitive. 5

6 3.2 SDA Algorithm for Solving the CAREs Assume that 0 solves the CARE (1.1). It is well-known that the CARE (1.1) can be rewritten as I I H = R (3.3) where H = A H G A T R = A G. The matrix H is a Hamiltonian matrix i.e. (HJ) T = HJ. Using Cayley transformation with some appropriate γ > 0 we can transform (3.3) into the following form I I M = L S (3.4) where M = H + γi L = H γi S = (R γi) 1 (R + γi). Now assume that we have chosen a γ > 0 such that the matrices A γ = A γi and W γ = A T γ + HA 1 γ G (3.5) are nonsingular. Chu et al. 7 proposed a method for computing γ such that both A γ and W γ are well-conditioned. Let A 1 γ 0 I A 1 T 1 = HA 1 γ GWγ T γ I 2 = 1 0 Wγ 1 (3.6) which are obtained by alternatively applying block Gaussian elimination to the matrices L and M (see 7 for more details). Then direct calculations gives rise to  0 I Ĝ M = T 2 T 1 M = Ĥ I L = T2 T 1 L = 0  T where  = I + 2γW T γ Ĝ = 2γA 1 γ GW 1 γ Ĥ = 2γW 1 γ HA 1 γ. Here the Sherman-Morrison-Woodbury formula is used. Since γ > 0 and HG 0 implies that ĜĤ 0 it follows that the resulting pencil M λ L is a SSF-1. In addition it follows from (3.4) that I M = L I S (3.7) Thus beginning with (3.7) following the same lines as Algorithm SDA-1 for solving the DARE we can construct a matrix sequence to approximate to the unique symmetric positive semi-definite solution to the CARE (1.1). For more details see Convergence Analysis of Algorithm SDA-1 Now we establish the convergence theory of Algorithm SDA-1 based on Theorem 2.1. The main results are listed in the following theorem. Theorem 3.1. Assume that Y 0 satisfy that = A T (I + G) 1 A + H (3.8) Y = AY (I + HY ) 1 A T + G (3.9) 6

7 where HG 0 that is assume that the DARE (3.8) and its dual DARE (3.9) have symmetric positive semi-definite solutions and Y respectively and let S = (I + G) 1 A T = (I + HY ) 1 A T. (3.10) Then the matrix sequences {A k } {G k } and {H k } generated by Algorithm SDA-1 satisfy that (a) A k = (I + G k )S 2k ; (b) H H k H k+1 and H k = (S T ) 2k ( + G k )S 2k (S T ) 2k ( + Y )S 2k ; (3.11) (c) G G k G k+1 Y and Y G k = (T T ) 2k (Y + Y H k Y )T 2k (T T ) 2k (Y + Y Y )T 2k. (3.12) Proof. Notice that UV 0 implies that I+UV is nonsingular and V (I+UV ) 1 (I+UV ) 1 U 0 it follows that Algorithm SDA-1 is well defined and Define H = H 0 H k H k+1 and G = G 0 G k G k+1. (3.13) M k = A k 0 H k I I Gk L k = 0 A T k Then the pencil M k+1 λl k+1 is a doubling transformation of the pencil M k λl k. Since (3.8) implies that I I M 0 = L 0 S (3.14) where S is defined by (3.10) applying (b) of Theorem 2.1 repeatedly we get I I M k = L k S 2k. (3.15) Equating the blocks of (3.15) yields that Combining (3.16) with (3.17) gives rise to A k = (I + G k )S 2k (3.16) H k = A T k S 2k. (3.17) H k = (S T ) 2k ( + G k )S 2k. (3.18) This together with (I + G k ) 0 implies that H k 0 i.e. H k. Similarly (3.9) can be rewritten as M 0 Y I where T is defined by (3.10) and from (3.19) we can derive that. Y T = L 0 (3.19) I Y G k = (T T ) 2k (Y + Y H k Y )T 2k which implies that Y G k. Thus the theorem is proved. 7

8 Let W = L I Y M Z = I I Y I. Noting that M 0 = M L 0 = L and Y 0 it follows from (3.14) and (3.19) that W and Z are nonsingular and satisfy W 1 S 0 MZ = W 1 I 0 LZ =. 0 I 0 T Thus by the spectral feature of a symplectic pencil it follows that if ρ(s) < 1 then it must have ρ(t) = ρ(s) < 1. In addition it is well known that 0 U V implies that U 2 V 2. Consequently from Theorem 3.1 we immediately get the following convergence result of Algorithm SDA-1. Corollary 3.2. Under the hypothesis of Theorem 3.1 if ρ(s) < 1 then we have (a) A k 2 (1 + 2 Y 2 ) S 2k 2 0 as k 0; (b) H k 2 + Y 2 S 2k as k 0; (c) Y G k 2 Y + Y Y 2 T 2k as k 0. Remark 3.1. The similar convergence result were obtained in 8. In contrast to the work of 8 however our analysis is much simpler and our convergence results are much stronger. Remark 3.2. Let G = BR 1 B T 0 with R > 0 and H = C T C 0 in the DARE (3.8) and assume that (AB) is stabilizable and (AC) is detectable. Then it is well known that the DARE (3.8) and its dual DARE (3.9) have symmetric positive semi-definite solutions and Y respectively and ρ(s) < 1 (see for example for details). Thus the conditions in Corollary 3.1 are satisfied. In fact it is easy to verify that if the DARE (3.8) and its dual DARE (3.9) have symmetric positive semi-definite solutions and Y respectively and ρ(s) < 1 with S = (I + G) 1 A then (AB) is stabilizable and (AC) is detectable. Remark 3.3. By Theorem 3.1 the matrix sequences {H k } and {G k } are monotonically increasing and bounded above and hence there exist symmetric positive semi-definite matrices H and Ḡ such that lim H k = H k lim G k = Ḡ. k Corollary 3.2 tells us that if ρ(s) < 1 then = H and Y = Ḡ. Now it is natural to ask if this is still true without the condition ρ(s) < 1. This is a very interesting and worthwhile problem. See the following two simple examples. Example 3.1. Let α1 0 ζ1 0 A = G = H = 0 α 2 0 ζ 2 θ where α 1 α 2 ζ 1 ζ 2 θ 1 R with ζ 2 0 ζ 1 θ 1 > 0 and α2 1 1+ζ 1 + θ 1 = 1. Direct calculation verifies 1 0 that with these given data the DARE (1.2) has = as its symmetric positive semi-definite 0 0 solution and in this case α1 S = (I G) 1 A = 1+ζ α 2 Since α 2 may be any real number ρ(s) can be any positive real number. Applying Algorithm SDA- 1 to these given data we have α (k) A k = 1 0 ζ (k) (k) 0 α (k) G k = 1 0 θ 2 0 ζ (k) H k =

9 where α (0) 1 = α 1 α (k+1) 1 = (α(k) 1 )2 1 + θ (k) α (0) 2 = α 2 α (k+1) 2 = (α (k) 2 )2 1 ζ(k) 1 ζ (0) 1 = ζ 1 ζ (k+1) 1 = ζ (k) 1 (1 + α(k+1) 1 ) ζ (0) 2 = ζ 2 ζ (k+1) 2 = ζ (k) 2 (1 + α(k+1) 2 ) θ (0) 1 = θ 1 θ (k+1) 1 = θ (k) 1 (1 + α(k+1) 1 ). From the proof of Theorem 3.1 we can see that in this case the equality H k = (S T ) 2k ( + G k )S 2k still holds and hence letting λ 1 = α1 1+ζ 1 we have 1 θ (k) 1 = (1 + ζ (k) 1 )λ2k 1 0 (k ) since λ 1 = α1 (k) 1+ζ 1 < 1 and 0 ζ 1 ζ1 θ 1 which can be proved by taking ζ 2 = 0 and using the same method as the proof of (c) of Theorem 3.1. This shows that in this case we still have lim k H k =. However when α 2 > 1 lim k α (k) 2 = +. Example 3.2. Let A = α1 0 G = 0 1 ζ1 0 H = 0 0 θ where α 1 ζ 1 θ 1 R with ζ 1 θ 1 > 0 and α2 1 1+ζ 1 + θ 1 = 1. It easy to verify that with these given 1 0 data for any ξ 0 the matrix ξ = 0 is a symmetric positive semi-definite solution of 0 ξ the DARE (1.2) which means that in this case the DARE (1.2) have infinitely many symmetric positive semi-definite solutions. Simple calculation gives rise to α1 S = (I G ξ ) 1 A = 1+ζ and so we have that ρ(s) = 1 for any ξ 0. We can easily prove that in this case we still have 1 0 ν 0 lim H k = 0 = lim k 0 0 G k = Y = k 0 0 where H k and G k are generated by applying Algorithm SDA-1 to these given data Y is a symmetric positive semi-definite solution of the dual DARE with ν = ζ1 θ 1. 4 SDA Algorithms for Preserving the SSF-2 In this section we shall first use the doubling transformations defined in the last section to derive two SDA algorithms for solving the NME-Ps and NME-Ms. Then we shall use the theory established in the last section to develop the convergence theory of these SDA algorithms. 4.1 SDA Algorithm for Solving the NME-Ps It is easy to verify that the NME-P (1.3) has a symmetric positive definite (i.e. > 0) if and only if satisfies I I M = L S (4.1) 9

10 for some matrix S R n n where A 0 M = Q I L = 0 I A T 0. (4.2) Notice that the pencil M λl is just a SSF-2. Therefore applying the special doubling transformation defined by (2.16) (2.18) repeatedly gives rise to the following structure-preserving doubling algorithm. Algorithm SDA-2. A 0 = A Q 0 = Q P 0 = 0 A k+1 = A k (Q k P k ) 1 A k Q k+1 = Q k A T k (Q k P k ) 1 A k P k+1 = P k + A k (Q k P k ) 1 A T k. Remark 4.1. Of course to ensure that this iteration is well defined the matrix Q k P k must be symmetric positive definite for all k. Below we shall prove that this condition can be guaranteed if the NME-P (1.3) has a symmetric positive solution (see Theorem 4.1). Remark 4.2. It is interesting to note that this algorithm is essentially the same as proposed in 26. In other words Meini s algorithm in 26 is a SDA algorithm. It has pointed out that this algorithm has very nice numerical behavior: having quadratical convergence rate low computational cost per step and good numerical stability. For more details see SDA Algorithm for Solving the NEM-Ms It is proved in 11 that there always exists a unique positive definite solution to the NME-M A T 1 A = Q (4.3) and moreover the spectral radius of 1 A is strictly less than 1. The solution is closely related the generalized eigenspace of the following pencil A 0 0 I M λl = λ Q I A T. (4.4) 0 In fact it is easy to verify that a symmetric positive definite matrix is a solution to the NME-M (4.3) if and only if satisfies that I I M = L S (4.5) for some matrix S R n n. Although the pencil (4.4) is not symplectic we can use the same technique as described in Section 2 to transform it into a symplectic pencil. Take then we have Direct calculation leads to M = AQ 1 0 A T Q 1 I  0 M 0 = M M = Q I L = I AQ 1 0 A T Q 1 M L = L M. (4.6) L0 = L L = P I  T 0 where  = AQ 1 A Q = Q + A T Q 1 A P = AQ 1 A T. (4.7) 10

11 Direct verification shows that M 0 λ L 0 is a symplectic pencil but neither a SSF-1 nor a SSF-2. Assume that > 0 is the unique symmetric positive solution to the NME-M (4.3). Then it satisfies the equality (4.5) with S = 1 A. Similar to the proof of (b) of Theorem 2.1 we can show that I M 0 = L I 0 S 2. (4.8) Now Let I I 0 = P I I Then it follows from (4.8) that M = I M Â 0 Q + P I L = 0 I Â T 0. I = L S 2. (4.9) Clearly now the pencil M λ L is a SSF-2. Thus beginning with (4.9) following the same lines as Algorithm SDA-2 for solving the NME-P (1.3) we can construct a matrix sequence to approximate to. Then the unique symmetric positive definite solution to the NME-M (4.3) can be obtained by computing = P. 4.3 Convergence Analysis of Algorithm SDA-2 Now we establish the convergence theory of Algorithm SDA-2 based on Theorem 2.1. The main results are listed in the following theorem. Theorem 4.1. Assume that > 0 satisfy that + A T 1 A = Q (4.10) where Q > 0 and let S = 1 A. Then the matrix sequences {A k } {Q k } and {P k } generated by Algorithm SDA-2 satisfy that (a) A k = ( P k )S 2k ; (b) 0 P k P k+1 < and (c) Q k+1 Q k Q and Proof. (Apply Mathematical Induction.) Denote Ak 0 M k = I Q k P k = ( P k ) + A T k ( P k ) 1 A k > 0; (4.11) Q k = (S T ) 2k ( P k )S 2k (S T ) 2k S 2k. (4.12) Q k L k = Pk I A T k 0 where P 0 = 0. For k = 1 since Q 0 P 0 = Q > 0 it follows that A 1 Q 1 P 1 are all well defined. Using the equality (4.10) we have A I 0 0 I A T = 1 A Q A T 1 > 0. (4.13) I 0 0 I Simple computation yields that I AQ 1 A I 0 0 I A T Q Q 1 A T = I Combining (4.14) and (4.13) we get AQ 1 A T 0. (4.14) 0 Q P 1 = AQ 1 A T > 0. (4.15) 11

12 On the other hand From (4.10) it is easy to verify that satisfies I I M 0 = L 0 S with S = 1 A. Since M 1 λl 1 is a doubling transformation of M 0 λl 0 Applying (b) of Theorem 2.1 we get I I M 1 = L 1 S 2. (4.16) Equating the blocks of (4.16) gives rise to This together with (4.15) implies that A 1 = ( P 1 )S 2 Q 1 = A T 1 S 2. Q 1 P 1 = ( P 1 ) + A T 1 ( P 1 ) 1 A 1 > 0 (4.17) Q 1 = (S T ) 2 ( P 1 )S 2 0. (4.18) Obviously the inequalities Q = Q 0 Q 1 and 0 = P 0 P 1 hold. Thus we have proved that the theorem is true for k = 1 Next assume that the theorem is true for all positive integers less or equal to k. Consider the case of k + 1. Since Q k P k > 0 it follows that A k+1 Q k+1 P k+1 are all well defined. Similar to the proof of (4.15) using the equality (4.11) we can prove that P k+1 = ( P k ) A k (Q k P k ) 1 A T k > 0. On the other hand since M j+1 λl j+1 is a doubling transformation of M j λl j for j = 01...k By using (b) of Theorem 2.1 k + 1 times we get I I M k+1 = L k+1 S 2k+1. (4.19) From (4.19) following the same lines as the proof of (4.17) and (4.18) it can be proved that Q k+1 P k+1 = ( P k+1 ) + A T k+1( P k+1 ) 1 A k+1 > 0 Q k+1 = (S T ) 2k+1 ( P k+1 )S 2k+1 0. Clearly P k P k+1 and Q k Q k+1 are true. This shows that the theorem is also true for integer k + 1. By induction principle the theorem is true for all positive integers. Remark 4.3. The similar results were obtained in 26 by using some properties of cyclic reduction and the spectral properties of block Toeplitz matrices having nonnegative definite matrix-valued generating functions. In contrast to the work of 26 however our analysis is much simpler and the results are much stronger. It was proved in 10 that if the matrix equation (1.3) has a symmetric positive definite solution then all symmetric solutions are positive definite and it has a maximal and minimal solution + and respectively. Since Theorem 4.1 is true for any symmetric positive definite solution to the NME-P (1.3) the following result follows immediately. Corollary 4.2. Under the hypothesis of Theorem 4.1 we have Q k P k > + 0 for all k where + and are the maximal and minimal solution of (1.3) respectively. In addition from Theorem 3.1 we immediately get the following result. 12

13 Corollary 4.3. Under the hypothesis of Theorem 4.1 if ρ(s) < 1 then we have (a) A k 2 2 S 2k 2 0 as k 0; (b) Q k 2 2 S 2k as k 0. Remark 4.4. By Theorem 4.1 the matrix sequence {Q k } is monotonically decreasing and bounded below by > 0 and hence there exists Q > 0 such that lim k Q k = Q. Corollary 4.3 tells us that if ρ(s) < 1 then = Q. In fact ρ(s) < 1 implies that is the maximal solution of (1.3) and moreover it has been proved that is the maximal solution of (1.3) if and only if ρ(s) 1 (see 16). Now assume that = + is the maximal solution of (1.3) it is natural to ask whether Q = + still holds if ρ(s) = 1. In 16 Guo proved that if ρ(s) = 1 and all eigenvalues of S on the unit circle are semisimple then Q = + is sill true and in this case the convergence is at least linear with rate 1/2. When S has nonsemisimple unimodular eigenvalues it is still open whether Q = +. Remark 4.5. It is proved that the NME-P (1.3) has a symmetric positive definite solution If and only if the nonlinear matrix equation Y + AY 1 A T = Q (4.20) has a symmetric positive solution Y (see for example 26). Assume that the maximal solution of (4.20) is Y +. Then it follows from (4.20) that A 0 I 0 I I T = Q I Q Y + A T (4.21) 0 Q Y + where T = Y 1 + A T. Similar to the proof of (4.12) from (4.21) we can show that 0 Q Y + P k = (T T ) 2k (Q k Q + Y + )T 2k (T T ) 2k Y + T 2k where P k and Q k are generated by Algorithm SDA-2. Since ρ(t) = ρ(y 1 + A T ) = ρ( 1 + A) (see for example 26) where + is the maximal solution of the NME-P (1.3) it follows that under the conditions of Corollary 4.3 we have lim k P k = Q Y +. If A is nonsingular then = Q Y + (see 26) where is the minimal solution of the NME-P (1.3) and so in this case we have lim k P k =. Remark 4.6. Since lim k (Q k P k ) = + if A is nonsingular and ρ(s) < 1 the under bound + in Corollary 4.2 is the best one. However + may be singular indeed it can be zero matrix. For example the NME-P (1.3) with Q = I and A = 1 2 I has + = = 1 2 I. 5 Conclusions In this paper we have introduced a structure-preserving transformation of a symplectic pencil referred to as the doubling transformation and developed its basic properties. Based on these nice properties of the transformation a unified convergence theory for the structure-preserving doubling algorithms for solving a class of Riccati-type matrix equations has been established by using only the knowledge from elementary matrix theory. References 1 G. Ammar and V. Mehrmann On Hamiltonian and symplectic Hessenberg forms Lin. Alg. Appl. 149 (1991) pp B. Anderson Second-order convergent algorithms for the steady-state Riccati equation Int. J. Control 28 (1978) pp W.N. Anderson T.D. Morley and G.E. Trapp Positive solutions to = A B 1 B Lin. Alg. Appl. 134 (1990) pp

14 4 Z. Bai J. Demmel and M. Gu An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems Numerische Mathematik 76 (1997) pp P. Benner and R. Byers Evaluating products of matrix pencils and collapsing matrix products Num. Lin. Alg. Appl. 8 (2001) pp E.K.-W. Chu H.-Y. Fan and W.-W. Lin A generalized structure-preserving doubling algorithm for generalized discrete-time algebraic Riccati equations preprint NCTS National Tsing Hua University Hsinchu 300 Taiwan E.K.-W. Chu H.-Y. Fan and W.-W. Lin A structure-preserving doubling algorithm for cotinuous-time algebraic Riccati equations to appear in Lin. Alg. Appl.. 8 E.K.-W. Chu H.-Y. Fan W.-W. Lin and C.-S. Wang A structure-preserving doubling algorithm for periodic descrete-time algebraic Riccati equations preprint NCTS National Tsing Hua University Hsinchu 300 Taiwan J.C. Engwerda On the existence of a positive definite solution of the matrix equation + A T 1 A = I Lin. Alg. Appl. 194 (1993) pp J.C. Engwerda A.C.M. Ran and A.L. Rijkeboer Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation + A 1 A = Q Lin. Alg. Appl. 186 (1993) pp A. Ferrante and B.C. Levy Hermitian solutions of the Equation = Q+N 1 N Lin. Alg. Appl. 247 (1996) pp G.H. Golub and C.F. Van Loan Matrix Computations 3rd ed. The Johns Hopkins University Press T. Gudmundsson C. Kenney and A. J. Laub Scaling of the discrete-time algebraic Riccati equation to enhance stability of the Schur solution method IEEE Trans. Auto. Control 37 (1992) pp C.-H. Guo Newton s method for discrete algebraic Riccati equations when the closed-loop matrix has eigenvalues on the unit circle SIAM J. Matrix Anal. Appl. 20 (1998) pp C.-H. Guo and P. Lancaster Iterative solution of two matrix equations Math. Comp. 68(1999) pp C.-H. Guo Comvergence rate of an iterative method for a nonlinear matrix equation SIAM J. Matrix Anal. Appl. 23 (2001) pp J.J. Hench and A.J. Laub Numerical solution of the discrete-time periodic Riccati equation IEEE Trans. Auto. Control 39 (1994) pp M. Kimura Convergence of the doubling algorithm for the discrete-time algebraic Riccati equation Int. J. Systems Science 19 (1988) pp D. Lainiotis N. Assimakis and S. Katsikas New doubling algorithm for the discrete periodic Riccati equation Applied Mathematics and Computation 60 (1994) pp P. Lancaster and L. Rodman Algebraic Riccati Equations Oxford University Press Oxford A.J. Laub A Schur method for solving algebraic Riccati equations IEEE Trans. Auto. Control 24 (1979) pp A.J. Laub Invariant subspace methods for numerical solution of Riccati equation in The Raccati Equation S. Bittanti A.J. Laub and J.C. Willems eds Springer-Verlag Berl 14

15 23 L.-Z. Lu and W.-W. Lin An iterative algorithm for the solution of the discrete time algebraic Riccati equations Lin. Alg. Appl. 189 (1993) pp L.-Z. Lu W.-W. Lin and C. E. M. Pearce An efficient algorithm for the discrete-time algebraic Riccati equation IEEE Trans. Auto. Control 44 (1999) pp V. Mehrmann The Autonomous Linear Quadratic Control Problem Lecture Notes in Control and Information Sciences 163 Springer-Verlag Berlin B. Meini Efficient computation of the extreme solutions of + A 1 A = Q and A 1 A = Q Math. Comp. 71 (2001) pp C. Paige and C. Van Loan A Schur decomposition for Hamiltonian matrices Lin. Alg. Appl. 41 (1981) pp T. Pappas A. J. Laub and N. R. Sandell On the numerical solution of the discrete-time algebraic Riccati equation IEEE Trans. Auto. Control 25 (1980) pp J.-G. Sun Sensitivity analysis of the discrete-time algebraic Riccati equation Lin. Alg. Appl. 275/276 (1998) pp J.-G. Sun and S.F. u Perturbation analysis of the maximal solution of the matrix equation + A T 1 A = P.II Lin. Alg. Appl. 362 (2003) pp P. Van Dooren A generalized eigenvalue approach for solving Riccati equations SIAM J. Sci. & Statistical Computing 2 (1981) pp S.F. u On the maximal solution of the matrix equation +A T 1 A = I Acta Scientiarum Naturalium Universitatis Pekinensis 36(2000) pp