structured doubling algorithms for weak hermitian...

STRUCTURED DOUBLING ALGORITHMS FOR WEAK

HERMITIAN SOLUTIONS OF ALGEBRAIC RICCATI

EQUATIONS

TSUNG-MIN HWANG AND WEN-WEI LIN

Abstract. In this paper, we propose structured doubling algorithms for the

computation of weak Hermitian solutions of continuous/discrete-time algebraic

Riccati equations. Under the assumptions that partial multiplicities of purely

imaginary and unimodular eigenvalues (if any) of the associated Hamiltonian

and symplectic pencil, respectively, are all even, we prove that the developedstructured doubling algorithms converge to the desired Hermitian solutions

globally and linearly. Numerical experiments show that structured doublingalgorithms perform efficiently and reliably.

1. Introduction

In this paper, we investigate structured doubling algorithms for the computationof Hermitian solutions X to(I) the continuous-time algebraic Riccati equation (CARE):

−XGX +AHX +XA+H = 0,(1.1)

or(II) the discrete-time algebraic Riccati equation (DARE):

X = AHX(I +GX)−1A+H,(1.2)

where A ∈ Cn×n, H = HH and G = GH = BBH ≥ 0 ∈ C

n×n with B ∈ Cn×m

being of full column rank, and I ≡ In denotes the identity matrix of order n.Furthermore, in CARE (1.1), the pair (A,B) is c-stablizable, i.e., if wHA = µwH

and wHB = 0 with w 6= 0, then Re(µ) < 0; in DARE (1.2), the pair (A,B) isd-stablizable, i.e., if wHA = µwH and wHB = 0 with w 6= 0, then |µ| < 1.

Equations (1.1) and (1.2) arise frequently in solving the unique “weak” sta-blizing controllers of continuous- and discrete-time H∞-optimal control systems,respectively [13, 15, 18, 25]. In addition, several applications in Wiener filteringtheory [47], network synthesis [2] and Moser-Veselov equations [9, 40] also need theHermitian solution of CAREs.

We consider the 2n× 2n Hamiltonian matrix H associated with the CARE:

H =

[A −G−H −AH

](1.3)

Date: November 13, 2006.

2000 Mathematics Subject Classification. 93B36, 93B40, 93B52, 93B60.Key words and phrases. algebraic Riccati equation, Hermitian solution, structured doubling

algorithm, purely imaginary eigenvalue, unimodular eigenvalue, global and linear convergence.This work is partially supported by the National Science Council and the National Center for

Theoretical Sciences of Taiwan.

1

2 TSUNG-MIN HWANG AND WEN-WEI LIN

which satisfies

HJ = −JHH , J =

[0 I

−I 0

];(1.4)

and consider the 2n × 2n symplectic pair (M,L) (or symplectic pencil M− λL)associated with the DARE:

M =

[A 0−H I

], L =

[I G

0 AH

](1.5)

which satisfies

MJMH = LJLH .(1.6)

Note that the special symplectic pair (M,L) of the forms in (1.5) is referred to asa standard symplectic form (SSF).

It is well-known that (e.g., [30, 32, 38, 41]) if[

A −G−H −AH

] [Ω1

Ω2

]=

[Ω1

Ω2

]Φ, Ω1,Ω2,Φ ∈ C

n×n,(1.7)

where the spectrum of Φ is contained in the closed left half plane, then X = Ω2Ω−11

(if Ω−11 exists) solves the CARE (1.1). In addition, if[

A 0−H I

] [Ω1

Ω2

]=

[I G

0 AH

] [Ω1

Ω2

]Φ, Ω1,Ω2,Φ ∈ C

n×n,(1.8)

where the spectrum of Φ is contained in the closed unit disk, then X = Ω2Ω−11

(if Ω−11 exists) solves the DARE (1.2). The particular invariant subspaces spanned

by[ΩH

1 , ΩH2

]Hin (1.7) and (1.8) are usually referred to as stable Lagrangian sub-

spaces.

Definition 1.1. A subspace U ⊆ C2n with dimension n is called a H-stable La-

grangian subspace, if U satisfies that (i) HU ⊆ U , (ii) U is isotropic, i.e., xHJy = 0,for all x, y ∈ U and (iii) Re(λ(H|U )) ≤ 0. Here λ(H|U ) denotes an eigenvalue of Hrestricted to U .

Definition 1.2. A subspace V ⊆ C2n with dimension n is called a (M,L)-stable

Lagrangian subspace, if (i) V is invariant under (M,L), i.e., there is a subspace Wsuch thatMV,LV ⊆ W, [44, pp. 303–305] (ii) V is isotropic and (iii) |λ((M,L)|V)| ≤1. Here λ((M,L)|V) denotes an eigenvalue of (M,L) restricted to V.

Unfortunately, anH-stable Lagrangian subspace and an (M,L)-stable Lagrangiansubspace do not always exist, when some purely imaginary eigenvalues of H andsome unimodular eigenvalues of (M,L) have odd partial multiplicities, respectively.Counterexamples can be found in [41]. To guarantee the existence of H- and(M,L)-stable Lagrangian subspaces, and the existence of Hermitian solutions ofCARE and DARE, respectively, throughout this paper, we assume that H in (1.3)and (M,L) in (1.5) satisfy the following assumptions.

(A1) The partial multiplicities (the sizes of Jordan blocks) of H associated withthe purely imaginary eigenvalues (if any) are all even.

(A2) The partial multiplicities of (M,L) associated with the unimodular eigen-values (if any) are all even.

We now describe essential equivalence statements for the existence of Hermitiansolutions of CARE and DARE, respectively.

STRUCTURED ALGORITHMS FOR RICCATI EQUATIONS 3

Theorem 1.1. [19, 29] Assume (A,B) in CARE (1.1) is c-stable. Then the fol-lowing statements are equivalent :

(i) The CARE (1.1) has a Hermitian solution;(ii) Assumption (A1) holds;(iii) There is an H-stable Lagrangian subspace.

The equivalence statements for the existence of Hermitian solutions of DAREhave been firstly given by [28] under three conditions related to a given matrix K sothat A−BK is stable and invertible, and ψK(η) > 0, for some |η| = 1 (see Theorem1.2 of [28] for details). In Theorem 1.2 we give different equivalence statements fora Hermitian solution of DARE under condition (1.9). These equivalence statementshave a correspondence to the statements in Theorem 1.1.

Theorem 1.2. Assume (A,B) in DARE (1.2) is d-stable. Suppose that there isan α ∈ C with |α| = 1 such that

λmin(BHA−Hα HA−1

α B) > −1(1.9)

where Aα = I − αA. Then the following statements are equivalent:

(i) The DARE (1.2) has a Hermitian solution;(ii) Assumption (A2) holds;(iii) There is an (M,L)-stable Lagrangian subspace.

Proof. See Appendix.For the continuous-time case, a well-known backward stable algorithm care, pro-

posed by [30], computes a Hermitian solution X for CARE by applying QR algo-rithm to H with reordering [43]. Unfortunately, the QR algorithm preserves neitherthe Hamiltonian structure nor the associated splitting eigenvalues. WhenH in (1.3)has no purely imaginary eigenvalues, a numerical strongly stable method has beenproposed by [10] for computing the Hamiltonian Schur form of H, and therefore,the H-stable Lagrangian subspace. Efficient structured doubling algorithms usingan appropriate Cayley transform [11, 27], as well as matrix sign function methods[5, 8, 14, 17] have been developed for solving the unique positive semidefinite so-lution of CARE (1.1). When H in (1.3) satisfies assumption (A1), an eigenvectordeflation technique proposed by [13] guarantees that the eigenvalues appear withthe correct pairing. This is certainly an advantages over the QR algorithm, but themethod ignores most of the structure of the problem during computation. A struc-tured algorithm proposed by [1] only using symplectic orthogonal transformations,computes the H-stable Lagrangian subspace. But there are numerical difficultiesin the convergence of the deflation steps when purely imaginary eigenvalues occur[38, p. 143]. To avoid the numerical difficulties mentioned above, another stableand structured algorithm has been developed in [33] as a preprocessing to deflateall purely imaginary eigenvalues and to get a reduced Hamiltonian matrix withno purely imaginary eigenvalues. When H satisfies (A1) with even=2, an efficientNewton’s method has been developed in [22] for solving CARE with globally linearconvergence.

For the discrete-time case, a well-known backward stable algorithm dare, pro-posed by [37, 42, 48], computes a Hermitian solution X for DARE by applying QZalgorithm to (M,L) with reordering. Unfortunately, this algorithm does not takeinto account the symplectic structure of (M,L). Non-structure-preserving itera-tive processes loosen the symplectic structure, this may cause that the algorithms


to fail or lose accuracy in adverse circumstances. When (M,L) in (1.5) has nounimodular eigenvalues, an efficient doubling algorithm was firstly derived in [3]based on an acceleration scheme of the fixed point iteration for (1.2). Quadraticconvergence of doubling algorithm has been shown in [26] and [35] with differentapproach. On the other hand, based on the viewpoint of the inverse-free iteration[4, 36], a matrix disk function method (MDFM) [6, 7] and a structure-preservingdoubling algorithm (SDA) [12, 24] have been developed for solving DARE. Bothof the symplectic structure in MDFM and SDA are preserved in SSF form at eachiterative step. However, the symplectic structure in MDFM can only be preservedin exact arithmetic. When (M,L) in (1.5) satisfies (A2), a structured algorithmhas been developed in [33] to deflate all unimodular eigenvalues as a preprocessingby the determination of their isotropic Jordan subbasis using the S+S−1-transformofM−λL [31]. When (M,L) satisfies (A2) with even=2, an efficient Newton-typemethod has been proposed by [21] to solve DARE with globally linear convergence.

As mentioned above, MDFM and SDA have been proposed for solving DARE(1.2) withM−λL having no unimodular eigenvalues. To solve CARE (1.1) with Hhaving no purely imaginary eigenvalues, the Hamiltonian matrix H is converted to

a symplectic pencil M − λL in SSF form by an appropriate Cayley transform andthen the MDFM method or the SDA algorithm can be applied. The main purposeof this paper is to apply MDFM or SDA for solving CAREs and DAREs, wherethe associated H in (1.3) andM− λL in (1.5) satisfy (A1) and (A2), respectively.Under these conditions, we prove the global linear convergence of MDFM and SDAalgorithms.

This paper is organized as follows. In Sections 2 and 3, we describe structureddoubling algorithms, D-MDFM/D-SDA and C-MDFM/C-SDA, for solving DAREsand CAREs, respectively. In Section 4, we prove that under assumptions (A1)and (A2), the structured doubling algorithms converge globally and linearly toHermitian solutions of DAREs and CAREs. In Section 5, we test several numericalexamples, illustrating the convergence behavior of MDFM, SDA and Newton-typemethods. Concluding remarks are given in Section 6.

Throughout this paper, we denote AH = AT the conjugate transpose of A ∈C

n×n, ı =√−1, and I ≡ In and 0 ≡ 0n the identity and zero matrices, respectively,

of order n. The jth column of In is denoted by ej . ‖ · ‖ denotes any matrix norm,σ(A) and ρ(A) denote the spectrum and the spectral radius of A, respectively.

2. Structured doubling algorithms for DAREs

The matrix disk function method (MDFM) in [6, 7] is developed to solve theDARE (1.2) by using a swapping technique built on the QR factorization. We referto this step as a QR-swap.

For a given symplectic pair (M,L) the QR-swap computes the QR-factorizationof [LT ,−MT ]T by

Q[L−M

]≡[Q11 Q12

M∗ L∗

] [L−M

]=

[R0

],(2.1)

where Q ∈ C4n×4n is unitary and R ∈ C

2n×2n is upper triangular. Let

L := L∗L, M :=M∗M.(2.2)


It is easily seen that (L,M) is symplectic. From (2.1)-(2.2) the pair (M, L) satisfiesthe doubling property,

Mx =M∗Mx = λM∗Lx = λL∗Mx = λ2L∗Lx = λ2Lx,(2.3)

provided that Mx = λLx.Algorithm 2.1 (D-MDFM for DAREs).

Input: A,G,H; τ (a small tolerance);Output: a Hermitian solution X to DARE.

Initialize: R ← 02n, M←[

A 0−H I

], L ←

[I G

0 AH

];

Repeat: Compute the QR-factorization:[Q11 Q12

M∗ L∗

] [L−M

]=

[R0

];

If ‖R − R‖ ≤ τ‖R‖, Then solve the leastsquared problem for X:

−M(:, 1 : n) =M(:, n+ 1 : 2n)X;Stop

Else

Set L ← L∗L, M←M∗M, R ← R;Go To Repeat

End If

The SDA algorithm in [12] is developed to solve DARE (1.2) with conditions ofstablizability and detectability by using a structured LU factorization instead ofthe QR factorization in (2.1). We refer to this step as a SLU -swap. As derived in[12], for a given pair (M,L) in SSF form (1.5), we construct

M∗ =

[A(I +GH)−1 0

−AH(I +HG)−1H I

], L∗ =

[I AG(I +HG)−1

0 AH(I +HG)−1

](2.4)

and consequently deduce that

M∗L = L∗M.(2.5)

We now compute L∗L and M∗M, and apply the Sherman-Morrison-Woodburyformula to produce

L =

[I G

0 AH

]= L∗L, M =

[A 0

−H I

]=M∗M,(2.6)

where

A = A(I +GH)−1A,(2.7a)

G = G+AG(I +HG)−1AH ,(2.7b)

H = H +AH(I +HG)−1HA.(2.7c)

Equations in (2.6) show that the matrix pair (M, L) is again in SSF form. From

(2.5)–(2.6), the pair (M, L) also satisfies the doubling property: Mx = λ2Lx.Remark 2.1.


(i) Equations in (2.7) have exactly the same form as the doubling algorithm(which has been first proposed and investigated in Anderson [3] and Kimura[26]). However, the original doubling algorithm was derived as an acceler-ation scheme for the fixed-point iteration from (1.2). Instead of producingthe sequence Xk, the doubling algorithm produces X2k. Furthermore,the convergence of the doubling algorithm was proven when A is nonsingu-lar [3], and for (A,G,H) which is reachable and detectable, or stablizableand observable [26]. A stronger convergent result of the SDA algorithmunder weak conditions (stablizability and detectability) can be found in[12, 35].

(ii) Note that when the assumption (A2) in Theorem 1.2 holds, the matrix

(I + GH) in (2.7) can be singular, thus A, G and H in (2.7) do not existand the SDA algorithm can break down. The numerical experience for thiscase shall be discussed in Section 5.

Algorithm 2.2 (D-SDA for DAREs).

Input: A,G,H; τ (a small tolerance);Output: a Hermitian solution X to DARE.

Repeat W ← I +GH;If W is singular or very ill-conditioned, then break down;Else solve for V1, V2 from WV1 = A, V2W

H = G;Set G← G+AV2A

H ,

H ← H + V H1 HA,

A← AV1

If ‖H −H‖ ≤ τ‖H‖, then X ← H, Stop.

Else set H ← H;End if

End if

Goto Repeat

3. Structured doubling algorithm for CAREs

To solve the CARE (1.1) a structured doubling algorithm was first proposedby Kimura [27] using Cayley transformation. With an appropriate γ > 0 theHamiltonian matrix H in (1.3) can be transformed to a symplectic pair (M,L) ≡(H + γI,H − γI) [38, 39], and then equivalently simplifies to a symplectic pair(M0,L0) in the SSF form. Here

M0 =

[A0 0−H0 I

], L0 =

[I G0

0 AH0

],(3.1)

with

A0 = I + 2γ(Aγ +GA−Hγ H)−1,(3.2a)

G0 = 2γA−1r G(AH

γ +HA−1γ G)−1,(3.2b)

H0 = 2γ(AHγ +HA−1

γ G)−1HA−1γ ,(3.2c)


and Aγ ≡ A− γI. The DARE associated with the symplectic pair (M0,L0) is

X = AH0 X(I +G0X)−1A0 +H0

on which Algorithm 2.1 or Algorithm 2.2 can then be applied. How to choose asuitable γ for Cayley transformation can be found in [11] for details.

Algorithm 3.1 (C-MDFM/C-SDA for CAREs).

Input: A,G,H; τ (a small tolerance);Output: a Hermitian solution X to CARE.(I) Find an appropriate value γ > 0 so that Aγ and

Aγ +GA−Hγ H are well-conditioned (see [11] for details).

(II) Initialize A0 ← I + 2γ(Aγ +GA−Hγ H)−1,

G0 ← 2γA−1r G(AH

γ +HA−1γ G)−1,

H0 ← 2γ(AHγ +HA−1

γ G)−1HA−1γ , k ← 0;

(III) Call Algorithm 2.1/Algorithm 2.2.

4. Convergence of structured doubling algorithms

Let

M =

[A 0−H I

], L =

[I G

0 AH

],(4.1)

where G = GH and H = HH . In light of the results in Theorem 1.2, we supposethat the matrix pair (M,L) is regular , i.e., det(M−λL) 6= 0, and satisfies (A2). Inthis section, we shall show that under these assumptions Algorithm 2.1 or Algorithm2.2 will converge to a weak Hermitian solution of DARE (1.2). A similar proof canbe applied to the convergence of Algorithm 3.1 to a weak Hermitian solution ofCARE when H satisfies (A1).

Denote the Jordan block of size p corresponding to a unimodular eigenvalueω ≡ eıθ by

Jω,p =

ω 1 0 · · · 0

0 ω 1. . .

......

. . .. . .

. . . 0...

. . .. . . 1

0 · · · · · · 0 ω

p×p

.(4.2)

We now prove some useful lemmas for the convergence theorem. The Jordanblock Jω,p to the power of 2k can be explicitly evaluated.

Lemma 4.1. [20, pp. 557] Let Jω,p be given by (4.2). Then

J2k

ω,p =

γ1,k γ2,k · · · γp,k

0 γ1,k

. . ....

.... . .

. . . γ2,k

0 · · · 0 γ1,k

,(4.3a)


where

γi,k =1

(i− 1)!

di−1

dxi−1x2k

∣∣∣∣x=ω

=2k(2k − 1) · · · (2k − i+ 2)

(i− 1)!ω2k−i+1,(4.3b)

for i = 1, . . . , p.

With p = 2m, let

Γk,m =

γm+1,k γm+2,k · · · γ2m−1,k γ2m,k

γm,k

. . .. . . γ2m−1,k

.... . .

. . .. . .

...

γ3,k

. . .. . . γm+2,k

γ2,k γ3,k · · · γm,k γm+1,k

m×m

(4.4)

≡ J2k

ω,2m(1 : m,m+ 1 : 2m),

in which γi,k are defined in (4.3b), for i = 2, . . . , 2m.To show that Γk,m is invertible, we first prove the following lemma.

Lemma 4.2. Let 2 ≤ r ≤ m and

Fr(m) =

f11 f12 · · · f1r

f21 f22 · · · f2r

......

...fr1 fr2 · · · frr

∈ R

r×r(4.5)

where

fij =

1, if j = 1,∏i+j−2

ν=i (m+ r − ν), if 2 ≤ j ≤ r,for i = 1, 2, . . . , r. Then

|det(Fr(m))| =r∏

ν=1

(ν − 1)!.(4.6)

Proof. Since

det(F2(m)) =

∣∣∣∣1 m+ 11 m

∣∣∣∣ = −1,

(4.6) is true for r = 2. Suppose that

|det(Fr−1(m))| =r−1∏

ν=1

(ν − 1)!.

Eliminating the first to (r−1)th entries in the first column of Fr(m) by elementaryrow operations, we obtain

Ir−2 0 0

0 1 −10 0 1

· · ·

1 −1 00 1 00 0 Ir−2

Fr(m)

=

[0 Fr−1(m)

1 m, m(m− 1), · · · , m(m− 1) · · · (m− r + 2)

],


where Fr−1(m) ≡ [fij ] ∈ R(r−1)×(r−1) with

fij =

1, if j = 1,

j∏i+j−2

ν=i [m+ (r − 1)− ν], if 2 ≤ j ≤ r − 1,

for i = 1, 2, . . . , r − 1. Using the factorization

Fr−1(m) = Fr−1(m)diag 1, 2, · · · , r − 1 ,we have

|det(Fr(m))| = (r − 1)! |det(Fr−1(m))| =r∏

ν=1

(ν − 1)!.

The proof is completed by mathematical induction.

Definition 4.1. Given ` ∈ Z. An m ×m Toeplitz matrix T = [tpq]m

p,q=1 can be

written in the form

T = [ti]2m+`−2i=`

with ti = tpq, where i = (m− 1)− (p− q) + ` and p, q = 1, . . . ,m.

Since |ω| = 1, for convenience, in the following discussion we w.l.o.g. assumethat ω = 1.

Lemma 4.3. The Toeplitz matrix Γk,m in (4.4) is nonsingular for k sufficiently

large and Γ−1k,m is Toeplitz in the big “O” sense having the form

Γ−1k,m =

1

O(2m2k

) [O(2ik)]m2−1

i=(m−1)2.(4.7)

Proof. The Toeplitz matrix T =[

1i!

]2m−1

i=1can be factorized by

T = diag

1

(2m− 1)!, · · · , 1

m!

Fm(m)Π,(4.8)

where Π = [en, · · · , e1] and Fm(m) is given by (4.5) with r = m. From Lemma 4.2,T is nonsingular. We write Γk,m of (4.4) in the form

Γk,m =

[2ik

i!+O

(2(i−1)k

)]2m−1

i=1

.(4.9)

By (4.8)–(4.9) we see that the dominant term of det(Γk,m) is det(T )2m2k withdet(T ) 6= 0. Thus, for sufficiently large k it holds det(Γk,m) 6= 0. From (4.9) followsthat

Γ−1k,m =

adj(Γk,m)

det(Γk,m)=

1

O(2m2k

) [O(2ik)]m2−1

i=(m−1)2.

Lemma 4.4. Let Jω ≡ Jω,m and Γk,m be defined in (4.2) and (4.4), respectively.Then

‖Γ−1k,mJ

2k

ω ‖ = O(2−k), ‖J2k

ω Γ−1k,mJ

2k

ω ‖ = O(2−k).(4.10)


Proof. From (4.3) we write the Toeplitz matrix J2k

ω by

J2k

ω = [γik]m+1i=−m+1 ,(4.11a)

where

γik =

0, for i = −m, . . . ,−1,O(2ik), for i = 0, 1, . . . ,m− 1.

(4.11b)

We compute Γ−1k,mJ

2k

ω and J2k

ω Γ−1k,mJ

2k

ω using (4.7) and (4.11) by the big “O” esti-mation. We then have

Γ−1k,mJ

2k

ω =1

O(2m2k

) [O(2ik)]m2−1

i=(m−1)2(4.12a)

and

J2k

ω Γ−1k,mJ

2k

ω =1

O(2m2k

) [O(2ik)]m2−1

i=(m−1)2.(4.12b)

This implies (4.10).

For the unimodular eigenvalues ωj = eıθj of (M,L) with even partial multiplicityp = 2mj , we have

Jωj ,2mj=

[Jωj ,mj

Γ1,mj

0mjJωj ,mj

], Γ1,mj

= emjeT1 ,(4.13)

for j = 1, . . . , r. From symplectic Kronecker’s Theorem for (M,L) (see [32]), there

exist a symplectic matrix Z (i.e., ZHJZ = J) and a nonsingular Q such that

QMZ =

[Js ⊕ J1 0` ⊕ Γ1

0n I` ⊕ J−H1

],(4.14a)

QLZ =

[I` ⊕ Iµ 0n

0n JHs ⊕ Iµ

],(4.14b)

where Js is the stable Jordan block of size ` (ρ(Js) < 1),

J1 = Jω1,m1⊕ · · · ⊕ Jωr,mr

,(4.15)

Γ1 = Γ1,m1⊕ · · · ⊕ Γ1,mr

with Γ1,mj= emj

eTmj

(j = 1, . . . , r), ` = n− (m1 + · · ·+mr) ≡ n−µ and ⊕ denotes

the direct sum of matrices. It is easily seen that spanZ(:, 1 : n) forms a uniquestable Lagrangian deflating subspace of (M,L) corresponding to Js⊕J1 (see [32]).

From (4.13), for each j = 1, . . . , r, there exists a suitable nonsingular Wj ∈C

mj×mj such that[Imj

00 W−1

j

] [Jωj ,mj

Γ1,mj

0 J−Hωj ,mj

] [Imj

00 Wj

]=

[Jωj ,mj

Γ1,mj

0 Jωj ,mj

].(4.16)

Let

Z = Z(In+` ⊕W1 ⊕ · · · ⊕Wr), Q = (In+` ⊕W−11 ⊕ · · · ⊕W−1

r )Q.(4.17)

Thus, from (4.16)-(4.17) equations (4.14a) and (4.14b), respectively, become

QMZ =

[Js ⊕ J1 0` ⊕ Γ1

0n I` ⊕ J1

]≡ JM,(4.18a)

QLZ =

[In 0n

0n JHs ⊕ Iµ

]≡ JL,(4.18b)


where Γ1 ≡ Γ1,m1⊕· · ·⊕Γ1,mr

with Γ1,mjbeing given in (4.13). It also follows that

spanZ(:, 1 : n) forms the unique stable Lagrangian deflating subspace of (M,L)corresponding to Js ⊕ J1.

Since JM and JL in (4.18) commute with each other and from (4.18), one canderive

MZJL = Q−1JLJM = LZJM.(4.19)

On the other hand, if we interchange the roles of M and L in (4.18) and considerthe symplectic pair (L,M), there are a nonsingular P and a symplectic Y such that

PLY = JM, PMY = JL,(4.20)

where spanY(:, 1 : n) forms the unique stable Lagrangian deflating subspace of(L,M) corresponding to Js ⊕ J1. Similar arguments also produce

LYJL =MY JM.(4.21)

Let (Mk,Lk)∞k=0 be the sequence of symplectic pairs generated by Algorithm 2.1,or (Mk,Lk)∞k=0 be the sequence of symplectic pairs in SSF with

Mk =

[Ak 0−Hk I

], Lk =

[I Gk

0 AHk

](4.22)

generated by Algorithm 2.2. With M0 = M,L0 = L, from (4.19) as well as(2.1)–(2.2) or (2.5)–(2.6) follows that

M1ZJ2L = M∗M0ZJ2

L =M∗L0ZJMJL = L∗M0ZJLJM(4.23)

= L∗L0ZJ2M = L1ZJ2

M.

By inductive process we have

MkZJ2k

L = LkZJ2k

M.(4.24)

By the result of Lemma 4.1 with p = 2mj and the definitions of Γk,m, JM and JLin (4.4), (4.18a) and (4.18b), respectively, equation (4.24) can be rewritten as

MkZ[In 0n

0n (JHs )2

k ⊕ Iµ

]= LkZ

[J2k

s ⊕ J2k

1 0` ⊕ Γk

0n I` ⊕ J2k

1

],(4.25)

where

Γk ≡ Γk,m1⊕ · · · ⊕ Γk,mr

(4.26)

with Γk,mjbeing defined in (4.4), for j = 1, . . . , r. Similarly, from (4.21) it also

holds

LkYJ2k

L =MkYJ2k

M.(4.27)

Lemma 4.5. Let J1 and Γk be defined in (4.15) and (4.26), respectively. Then Γk

is invertible and satisfies

‖Γ−1k J2k

1 ‖ = O(2−k), ‖J2k

1 Γ−1k J2k

1 ‖ = O(2−k).(4.28)

Proof. By definitions the lemma follows from Lemma 4.4 immediately.


Theorem 4.1. Let (M,L) be given in (1.5) satisfying (A2) and the sequence(Mk,Lk) be generated by Algorithm 2.1. Let Z satisfy (4.19) and be partitionedby

Z =

[Z1 Z3

Z2 Z4

],(4.29)

where Zi ∈ Cn×n for i = 1, . . . , 4. Suppose that Z1 is invertible. Then

‖Mk(:, 1 : n) +Mk(:, n+ 1 : 2n)X‖(4.30)

≤ O(ρ(Js)

2k)

+O(2−k

)→ 0, as k →∞,

where X ≡ Z2Z−11 solves DARE (1.2).

Proof. Since span[ZT1 , Z

T2 ]T forms the unique stable Lagrangian subspace of

(M,L) corresponding to Js⊕J1, X ≡ Z2Z−11 is Hermitian and solves DARE (1.2).

To show (4.30) we partition Mk,Lk conformally with (4.29) into

Mk =

[Mk,1 Mk,3

Mk,2 Mk,4

], Lk =

[Lk,1 Lk,3

Lk,2 Lk,4

].(4.31)

Substituting (4.29) and (4.31) into (4.25) we have

Mk,1Z1 +Mk,3Z2 = Lk,1Z1

(J2k

s ⊕ J2k

1

)+ Lk,3Z2

(J2k

s ⊕ J2k

1

),(4.32a)

(Mk,1Z3 +Mk,3Z4)((JH

s )2k ⊕ Iµ

)= Lk,1

[Z1 (0` ⊕ Γk) + Z3

(I` ⊕ J2k

1

)]

+Lk,3

[Z2 (0` ⊕ Γk) + Z4

(I` ⊕ J2k

1

)],(4.32b)

Mk,2Z1 +Mk,4Z2 = Lk,2Z1

(J2k

s ⊕ J2k

1

)+ Lk,4Z2

(J2k

s ⊕ J2k

1

),(4.32c)

(Mk,2Z3 +Mk,4Z4)((JH

s )2k ⊕ Iµ

)= Lk,2

[Z1 (0` ⊕ Γk) + Z3

(I` ⊕ J2k

1

)]

+Lk,4

[Z2 (0` ⊕ Γk) + Z4

(I` ⊕ J2k

1

)].(4.32d)

Postmultiplying (4.32b) by(0` ⊕ Γ−1

k J2k

1

)Z−1

1 to eliminate Lk,1Z1

(0` ⊕ J2k

1

)and

Lk,3Z2

(0` ⊕ J2k

1

)in (4.32a) we get

Mk,1 +Mk,3X = (Mk,1Z3 +Mk,3Z4)(0` ⊕ Γ−1

k J2k

1

)Z−1

1(4.33)

− (Lk,1Z3 + Lk,3Z4)(0` ⊕ J2k

1 Γ−1k J2k

1

)Z−1

1

+(Lk,1Z1 + Lk,3Z2)(J2k

s ⊕ 0µ

)Z−1

1 .

Similarly, from (4.32c) and (4.32d) follows that

Mk,2 +Mk,4X = (Mk,2Z3 +Mk,4Z4)(0` ⊕ Γ−1

k J2k

1

)Z−1

1(4.34)

− (Lk,2Z3 + Lk,4Z4)(0` ⊕ J2k

1 Γ−1k J2k

1

)Z−1

1

+(Lk,2Z1 + Lk,4Z2)(J2k

s ⊕ 0µ

)Z−1

1 .

Since ‖Mk,i‖ and ‖Lk,i‖ are bounded, for i = 1, . . . , 4, from (4.33)–(4.34) andLemma 4.5 follows the assertion (4.30).


Theorem 4.2. Let (M,L) be given in (1.5) satisfying (A2). Suppose the sequenceAk, Gk, Hk generated by Algorithm 2.2 is well-defined and

‖Gk‖ ≤ O(2k), for sufficiently large k.(4.35)

Let Z and Y satisfy (4.19) and (4.21), respectively, and be partitioned by

Z =

[Z1 Z3

Z2 Z4

], Y =

[Y1 Y3

Y2 Y4

],(4.36)

where Zi, Yi ∈ Cn×n, for i = 1, . . . , 4. Suppose that Z1 is invertible. Then

(a) if ‖Gk‖ = O(2k)→∞, as k →∞, it holds that(i) ‖Ak‖ ≤ O(1) bounded,

(ii) ‖X −Hk‖ ≤ O(ρ(Js)

2k)

+O(2−k)→ 0 as k →∞,

where X = Z2Z−11 solves the DARE (1.2);

(b) if ‖Gk‖ ≤ O(1), for all k, it holds that(i) ‖Ak‖ ≤ O(2−k)→ 0, as k →∞,

(ii) ‖X −Hk‖ ≤ O(ρ(Js)

2k)

+O(2−k)→ 0, as k →∞,

(iii) if Y2 is invertible, then ‖Y − Gk‖ ≤ O(ρ(Js)

2k)

+ O(2−k) → 0, as

k →∞, where Y = −Y1Y−12 solves the dual DARE

Y = AY (I +HY )−1AT +G,

(iv) I +GkHk → singular matrix as k →∞.

Proof. . From (4.18), [ZT1 , Z

T2 ]T spans the unique stable Lagrangian subspace of

(M,L) corresponding to Js ⊕ J1. So, X ≡ Z2Z−11 is Hermitian and solves DARE

(1.2). Substituting (Mk,Lk) of (4.22) and Z of (4.36) into (4.25), and comparingboth sides we obtain

AkZ1 = (Z1 +GkZ2)(J2k

s ⊕ J2k

1

),(4.37a)

AkZ3

((JH

s )2k ⊕ Iµ

)= (Z1 +GkZ2) (0` ⊕ Γk)(4.37b)

+(Z3 +GkZ4)(I` ⊕ J2k

1

),

−HkZ1 + Z2 = ATk Z2

(J2k

s ⊕ J2k

1

),(4.37c)

(−HkZ3 + Z4)((JH

s )2k ⊕ Iµ

)= AT

k Z2 (0` ⊕ Γk) +ATk Z4

(I` ⊕ J2k

1

).(4.37d)

Claim (a) (i) and (b) (i): Postmultiplying (4.37b) by(0` ⊕ Γ−1

k J2k

1

)Z−1

1 and

using (4.37a) we have

Ak

[I − Z3

(0` ⊕ Γ−1

k J2k

1

)Z−1

1

]= (Z1 +GkZ2)

(J2k

s ⊕ 0µ

)Z−1

1(4.38)

− (Z3 +GkZ4)(0` ⊕ J2k

1 Γ−1k J2k

1

)Z−1

1 .

By Lemma 4.5 and assumptions in (a) or (b), the sequence Ak satisfies

‖Ak‖ ≤ O(2k) ·O(ρ(Js)

2k)

+O(2k) ·O(2−k) = O(1)(4.39)

for case (a); or

‖Ak‖ ≤ O(1) ·O(ρ(Js)

2k)

+O(1) ·O(2−k)→ 0,(4.40)


as k →∞, for case (b).

Claim (a) (ii) and (b) (ii): Postmultiplying (4.37d) by(0` ⊕ Γ−1

k J2k

1

)Z−1

1 and

using (4.37c), we get

−Hk

[I − Z3

(0` ⊕ Γ−1

k J2k

1

)Z−1

1

]+X(4.41)

= Z4

(0` ⊕ Γ−1

k J2k

1

)Z−1

1 −ATk Z4

(0` ⊕ J2k

1 Γ−1k J2k

1

)Z−1

1

+ATk Z2

(J2k

s ⊕ 0µ

)Z−1

1 .

By Lemma 4.5 and the boundedness of Ak in (4.39) or (4.40), the matrix X−Hk

in (4.41) can be bounded by

‖X −Hk‖ ≤ O(ρ(Js)

2k)

+O(2−k),

for k sufficiently large.Claim (b) (iii): Substituting (Lk,Mk) of (4.22) and Y of (4.36) into (4.27) we

have

Y1 +GkY2 = AkY1

(J2k

s ⊕ J2k

1

),(4.42a)

(Y3 +GkY4)((JT

s )2k ⊕ Iµ

)= AkY1

(0` ⊕ Γk

)+AkY3

(I` ⊕ J2k

1

).(4.42b)

As above, postmultiplying (4.42b) by(0` ⊕ Γ−1

k J2k

1

)Y −1

2 and using (4.42a), we get

−Y +Gk

[I − Y4

(0` ⊕ Γ−1

k J2k

1

)Y −1

2

](4.43)

= Y3

(0` ⊕ Γ−1

k J2k

1

)Y −1

2 −AkY3

(0` ⊕ J2k

1 Γ−1k J2k

1

)Y −1

2

+AkY1

(J2k

s ⊕ 0µ

)Y −1

2 .

By Lemma 4.5 and boundedness of Ak we show that

‖Y −Gk‖ ≤ O(ρ(Js)

2k)

+O(2−k),

for k sufficiently large.Claim (b) (iv): From (4.37a) and (4.37c), we have

AkZ1 = (I +GkX)Z1

(J2k

s ⊕ J2k

1

),(4.44)

−Hk +X = ATk Z2

(J2k

s ⊕ J2k

1

)Z−1

1 .(4.45)

Multiplying (4.45) by Gk, we get

−(I +GkHk) + (I +GkX) = GkATk Z2

(J2k

s ⊕ J2k

1

)Z−1

1

which implies that

(I +GkHk)Z1 − (I +GkX)Z1 = −GkATk Z2

(J2k

s ⊕ J2k

1

).(4.46)

Postmultiplying (4.46) by(J2k

s ⊕ J2k

1

)and using the result in (4.44), then

(I +GkHk)Z1

(J2k

s ⊕ J2k

1

)= AkZ1 −GkA

Tk Z2

(J22k

s ⊕ J22k

1

).(4.47)


SDA [12] MDFM [7] NTM [23]

Flops 233 n

3 3523 n3 30n3

Table 1. Flop counts in each iteration.

Postmultiply (4.47) by(0` ⊕ J−2k

1

) [0 Iµ

]Tto get

(I +GkHk)Z1

[0Iµ

]= AkZ1

[0

J−2k

1

]−GkA

Tk Z2

[0

J2k

1

].(4.48)

Using the assumption ‖Gk‖ ≤ O(1) for all k and the result of (4.40), the firstcolumn of (4.48) becomes

(I +GkHk)Z1

[0e1

]= AkZ1

[0

ω−2k

1 e1

]−GkA

Tk Z2

[0

ω2k

1 e1

]→ 0, as k →∞.

Therefore, I +GkHk converges to a singular matrix as k →∞.

Note that in Section 5 we show examples which satisfy either the assumption of‖Gk‖ in (a) or (b) of Theorem 4.2.

5. Numerical results

In this section, we test some numerical examples satisfying assumptions (A1)and (A2), respectively, on the SDA algorithms for DAREs and CAREs to illustratethe convergence behavior. All computations were performed in MATLAB R2006aon a PC with an Intel Pentium-IV 3.4 GHz processor and 2 GB main memory,using IEEE double-precision floating-point arithmetic (eps ≈ 2.22 × 10−16). Theoperating system running the machine is Fedora Core 2 with Kernel 2.6.10-1.771-FC2.

The MATLAB commands “dare” and “care” fail in the initial step becauseone of the matrices G and H in (1.1) or (1.2) is indefinite. Furthermore, thestrongly stable method [10] and matrix sign function methods [5, 8, 14, 17] fail toconverge because the associated Hamiltonian matrix has purely imaginary eigen-values. Therefore, in our numerical testing, we compare SDA algorithms [11, 12]with MDFM methods [6, 7] and Newton’s methods [21, 22]. We summarize the flopcounts for each iteration in SDAs, MDFMs and NTMs in Table 1.

For tables in the following examples, data for various methods are listed incolumns with obvious headings. The heading “D-SDA”, “D-MDFM” and “D-NTM” stand for Algorithm 2.2, Algorithm 2.1 and Newton’s method [21] applied toDARE, respectively. “C-SDA” and “C-MDFM” stand for Algorithm 3.1 while call-ing Algorithm 2.2 and 2.1, respectively, in Step (III). “C-NTM” stands for Newton’smethod [22] applied to CARE.

We report the numbers of iterations by “ITs”, the total flops (= Flops × ITs)by “TFs” for algorithms, and the maximal error between the accurate and theapproximate stable eigenvalues of (I + GX)−1A or A − GX by “Err”, where Xis an approximate solution to DARE or CARE, respectively. Let λi be the exactstable eigenvalue and λi be the corresponding approximate eigenvalue. The “Err”


is defined by

Err = max1≤i≤n

|λi − λi|.

The algorithm is terminated when the residual of DARE or CARE cannot be re-duced further.

5.1. Discrete-time algebraic Riccati equations. In this subsection, we shallreport the numerical comparison of D-SDA, D-MDFM and D-NTM. The conver-gence of D-NTM is globally linear [21] when the size of Jω,p in (4.2) is two andthe matrix G is positive semidefinite. Furthermore, it needs to choose an initialmatrix L0 so that A − BL0 is d-stable. If p > 2, the convergence of D-NTM cannot be guaranteed. In D-SDA and D-MDFM, G and H are only required to besymmetric and p is an even number. In the following, we shall show three examplesto compare the performance of D-SDA, D-MDFM and D-NTM. The matrix G inthe first example is negative definite. Thus, D-NTM can not be applied. The sizeof Jordan block Jω,p in the second example is two, so the linear convergence ofD-NTM is guaranteed. The other example with p > 2 the convergence of D-NTMis numerically linear.

For the residual of DARE, we use the “normalized” residual (DNRes) formula

DNRes ≡ ‖AT X(I +GX)−1A+H − X‖‖X‖+ ‖AT X(I +GX)−1A‖+ ‖H‖

,

proposed in [7], where X is an approximate solution to DARE.

Example 5.1. [34, Example 2.2] Let

A =

[0 −3+

√5

23−

√5

2 0

], G = −5

(√5− 1

2

)2

I2 and H = I2.

The symplectic pencil (M,L) has eigenvalues ı, ı,−ı,−ı and is equivalent to([ı 10 ı

]⊕ I2, I2 ⊕

[ı 10 ı

]), because rank(M± ıL) = 3. On the other hand,

we have

M[

I22

5−√

5I2

]= L

[I2

25−

√5I2

] [0 1−1 0

]

and [AT 0−G I

] [I2√

5−52 I2

]=

[I H

0 A

] [I2√

5−52 I2

] [0 −11 0

].

These imply that the unique Hermitian solutions of the DARE and the dual DAREare

X∗ =

(2

5−√

5

)I2, Y∗ =

(√5− 5

2

)I2,(5.1)

respectively. It holds that I + Y∗X∗ = 0.

The numerical results by using D-SDA and D-MDFM for computing X arereported in Table 2. The notation “∗” in the fourth column of Table 2 means thatD-NTM cannot be applied, since G is not positive semidefinite. We see that thenormalized residual, i.e., the backward error, for X by D-SDA has 16 significant


D-SDA D-MDFM D-NTM

DNRes 5.55× 10−17 5.68× 10−12 ∗ITs 25 18 ∗TFs(n3) 192 2112 ∗Err 2.48× 10−8 5.72× 10−6 ∗

Table 2. Results for Example 5.1.

digits and attains the machine accuracy. It has only 12 significant digits by D-MDFM.

Compared to the DNRes which has the machine accuracy, the forward errorsof stable eigenvalues have only eight and six significant digits for D-SDA and D-MDFM, respectively. This is due to the poor separation between the d-stable andd-unstable subspectra of (M,L) [45, 46]. In fact, one can easily check that thesymplectic pencil

[0 0−H∞ I

]− λ

[I G∞0 0

]

is singular if and only if I+G∞H∞ is singular. Then from the fact that I+Y∗X∗ = 0,it follows that the pencilMk − λLk generated by the D-SDA algorithm tends to asingular pencil as k →∞.


M0 =

[A0 00 I8

]and L0 =

[I8 G0

0 AT0

]

with

A0 =

−1

11

⊕

[ √3

212

− 12

√3

2

]⊕

12 1

12 1

12

and

G0 =

11 1

. . .. . .

1 1

11 1

. . .. . .

1 1

T

∈ R8×8.

The symplectic pencil (M0,L0) has eigenvalues −1, 1,√

3−ı2 ,

√3+ı2 , 2, 1

2 with par-tial multiplicities 2, 2, 2, 2, 3, 3. For the corresponding DARE, all of the matricesHk generated by D-SDA are equal to zero matrix which is the Hermitian solutionof DARE. That is, D-SDA only needs one iteration to obtain exact solution. There-fore, for such symplectic pencil (M0,L0), the performance of D-SDA is obviouslybetter than D-MDFM and D-NTM.

In order to get a fair comparison of these three methods, we take an equivalenttransformation for (M0,L0). Set the “state” in the MATLAB command “randn”to be [3616447672, 4257789242]T and the matrix H0 to be a diagonal matrix with


D-SDA D-MDFM D-NTM

DNRes 4.54× 10−15 1.39× 10−12 1.08× 10−16

ITs 24 20 20TFs(n3) 184 2347 600Err 7.14× 10−6 1.17× 10−5 7.23× 10−6


normally distributed random diagonal elements so that I − G0H0 is nonsingular.Define a new equivalent symplectic pencil (M,L) as

M =

[A 0−H I

]≡

[(I −G0H0)

−1 00 I

] [I 0

AT0 H0(I −G0H0)

−1 I

]

×[A0 00 I

] [I 0−H0 I

](5.2a)

and

L =

[I G

0 AT

]≡

[(I −G0H0)

−1 00 I

] [I 0

AT0 H0(I −G0H0)

−1 I

]

×[I G0

0 AT0

] [I 0−H0 I

].(5.2b)

Note that G and H in (5.2) are positive definite and indefinite, respectively.The matrix L0 in D-NTM is taken as a normally distributed random diagonal

matrix with state = [3080845898, 338958098]T . The numerical results by using

D-SDA, D-MDFM and D-NTM for computing X of DARE with matrices A, G andH in (5.2) are reported in Table 3.

Example 5.3. Let

M0 =

[A0 00 I7

]and L0 =

[I7 G0

0 AT0

]

with

A0 =

[ −1 − 12

−1

]⊕

−1 − 1

2 − 18

−1 − 12

−1

⊕

(−1

3I2

)

and

G0 =

[ 132

18

18

12

]⊕

1512

1128

132

1128

132

18

132

18

12

⊕

(2

9I2

).

The symplectic pencil (M0,L0) has eigenvalues −3,−3, 13 ,

13 ,−1,−1 and eigen-

value -1 has partial multiplicities 4, 6.Taking H0 = −3.5I7 and using the equivalent transformation as in (5.2), we get

a new DARE with G and H being positive semidefinite and indefinite, respectively.The numerical results by using D-SDA, D-MDFM and D-NTM for computing X of


D-SDA D-MDFM D-NTM

DNRes 3.76× 10−13 2.03× 10−10 1.52× 10−10

ITs 18 18 31TFs(n3) 138 2112 930Err 1.80× 10−3 1.49× 10−3 1.78× 10−3


this DARE are reported in Table 4. Here, the matrix L0 in D-NTM is a normallydistributed random diagonal matrix with state = [2355717396, 3700125409]T .

From Table 4, we see that the backward error for X by D-SDA has 13 significantdigits which is better than that by D-MDFM and D-NTM. The forward errors ofstable eigenvalues by using these three methods are almost equal to 6

√eps.

5.2. Continuous-time algebraic Riccati equations. In this subsection, weshall report the numerical comparison of C-SDA, C-MDFM and C-NTM by threeexamples. Note that, similar to D-NTM, C-NTM converges globally and linearlywhen the size of Jω,p in (4.2) is two. If p > 2, the convergence of C-NTM cannotbe guaranteed. Consequently, we give three examples to illustrate the numericalbehavior. In Example 5.4, C-NTM is guaranteed to converge linearly, where p = 2.In Example 5.5, where the maximal size of Jω,p is four, the convergence of C-NTMis numerically linear. However, in Example 5.6, where the maximal size of Jω,p

is eight, the sequence generated by C-NTM is divergent. Note that in C-NTM itrequires to choose an initial matrix X0 so that A−GX0 is stable.

For the residuals of CARE, we use the “normalized” residual (CNRes) formula

CNRes ≡ ‖ − XGX +AT X + XA+H‖‖XGX‖+ ‖AT X‖+ ‖XA‖+ ‖H‖

,

proposed in [11], where X is an approximate solution to CARE.


H0 =

[A0 −G0

0 −AT0

]

where

A0 = 02 ⊕[

0 1−1 0

]⊕[

0 2−2 0

]⊕[−1 10 −1

]

and

G0 =

2 1 1

1 2. . .

. . .. . . 1

1 1 2

∈ R

8×8.

The eigenvalues of H0 are 1,−1, 0, 0, ı,−ı, 2ı,−2ı and the partial multiplicity ofeach eigenvalue is two. For this CARE, each of Hk in C-SDA is a zero matrix whichis the Hermitian solution of CARE. Thus, the exact solution of CARE is obtainedafter one iteration by C-SDA.


C-SDA C-MDFM C-NTM

CNRes 3.78× 10−16 5.39× 10−14 2.08× 10−16

ITs 26 23 30TFs(n3) 199 2699 900Err 2.27× 10−8 3.58× 10−7 3.75× 10−8


Now, we use the following similarity transformation to get a new Hamiltonianmatrix H:

H =

[A −G−H −AT

]≡[

12I

√3

2 I

−√

32 I

12I

] [A0 −G0

0 −AT0

] [12I −

√3

2 I√3

2 I12I

].

The numerical results by using C-SDA, C-MDFM and C-NTM for computing X

of the new CARE are reported in Table 5. Here, the initial matrixX0 in C-NTM is anormally distributed random diagonal matrix with state = [549776141, 3491173770]T

and we take γ = 2.2 in the Cayley transformation for C-SDA and C-MDFM.From Table 5, it shows that the normalized residuals for X by C-SDA and

C-NTM have 16 significant digits which are better than that by C-MDFM. Theaccuracy for stable eigenvalues by C-SDA and C-NTM are also better than that byC-MDFM.

Example 5.5. Let A0 and G0 be 5× 5 real matrices defined by

A0 =

U I2 00 U 00 0 a

, G0 =

0 0 00 I2 00 0 g

where a, g are parameters and U =

[0 β

−β 0

], β 6= 0. The matrix

[A0 −G0

0 −AT0

]

has nonzero eigenvalues a,−a and pure imaginary eigenvalues βı,−βı with partialmultiplicity 4.

Define[A −G−H −AT

]=

[PT 00 PT

] [C S

−S C

] [A0 −G0

0 −AT0

] [C −SS C

] [P 00 P

],

where P = I5 − 2uuT is a Householder matrix and

C =

[I4 00 c

], S =

[04 00 s

]

with c > 0, s > 0 and c2 + s2 = 1. Taking

a = 0.6781616521431886, β = 5.513985806849778,

g = 43.14437852853182, c = 0.3559455724227920,

u =

0.82103737884154660.43654443788074480.16512831562542100.3286639287858224

0.6263991871530291× 10−2

,


C-SDA C-MDFM C-NTM

CNRes 1.00× 10−16 3.25× 10−9 8.40× 10−13

ITs 9 17 34TFs(n3) 69 1995 1020Err 3.27× 10−8 5.08× 10−4 2.45× 10−5


we have that G and H are positive and negative semidefinite, respectively.Taking γ = 35 in the Cayley transformation and the initial X0 in C-NTM to be

a normally distributed random matrix with state = [2406827087, 4217562603]T ,

numerical results by using C-SDA, C-MDFM and C-NTM for computing X arereported in Table 6. It shows that the CNRes for X by C-SDA attains the machineaccuracy. C-MDFM and C-NTM have only 9 and 13 significant digits, respectively.Furthermore, the accuracy of stable eigenvalues by C-SDA is better than that byC-MDFM and C-NTM.

Example 5.6. [33] Let

A0 = 0⊕[

0 01 0

]⊕

0 0 0 01 0 0 00 1 0 00 0 1 0

⊕ (−I2),

H0 = (−1)⊕[

0 00 −1

]⊕

0 0 0 00 0 0 00 0 0 00 0 0 −1

⊕ (−I2).

Construct a Hamiltonian matrix H in exact arithmetic:

H =

[I V2

0 I

] [V T

1 00 V −1

1

] [A0 0H0 −AT

0

] [V −T

1 00 V1

] [I −V2

0 I

],

where

V1 =

1 1 0. . .

. . .

. . . 10 1

, V2 =

1 1 01 −1 2

2 1 3. . .

. . .. . .

7 −1 80 8 1

.

It is easily seen that H has nonzero eigenvalues −1,−1, 1, 1 and the zero eigenvaluehas partial multiplicities 2, 4, 8.

Taking γ = 10.5 in the Cayley transformation and the initial X0 in C-NTM tobe a normally distributed random matrix with state = [2902100237, 857523738]T ,numerical results by using C-SDA, C-MDFM and C-NTM are reported in Table7. It shows that the normalized residual for X by C-SDA and C-MDFM has 13and 9 significant digits, respectively, while the C-NTM is divergent. The CNResby C-SDA lost 3 significant digits compared to the machine accuracy, because the


C-SDA C-MDFM C-NTM

CNRes 7.51× 10−13 6.07× 10−9 ∗ITs 20 17 ∗TFs(n3) 153 1995 ∗Err 6.56× 10−2 5.55× 10−2 ∗


highest partial multiplicity of the zero eigenvalue of H is eight, making it verysensitive to perturbation [45, 46]. The forward errors of stable eigenvalues by usingC-SDA and C-MDFM are attained to 8

√eps.

5.3. Comments.

(i) For SDAs, MDFMs and NTMs, it requires 233 n

3 [12], 3523 n3 [7] and 30n3 [23]

flops, respectively, in each iterative step. The flops count of SDA is about7% and 26% of that of MDFM and NTM, respectively. This is mainly dueto the fact that the main step in MDFM involves the QR-factorization of[LT ,−MT

]Tand the formation of Q ∈ C

4n×4n, all in higher dimensions.The operations in SDA are all in C

n×n while keeping the SSF form.(ii) Examples investigated here are all ill-conditioned in solving DARE or CARE

because the associated symplectic pencils or Hamiltonian matrices haveeigenvalues on the unit circle or on the purely imaginary axis. However,the SDA algorithms solve them efficiently and accurately without failureand with less flops count.

(iii) Tables 2, 5 and 6 show that the approximate solutions X from SDA areas accurate as the machine accuracy, and more accurate than those fromMDFM. This behavior illustrates the importance of the SSF property, andthe consequent global and linear convergence.

(iv) For Examples 5.1, 5.2, 5.3, 5.4 and 5.6, the convergence behavior of Ak,Gk and Hk coincides with the result of Theorem 4.2 (b) which convergeslinearly and the matrix I+GkHk tends to a singular matrix as k →∞. Weuse Example 5.1 to illustrate such convergence behavior. Let X∗ and Y∗ bedefined in (5.1), then the Frobenius norms of Ak, Gk − Y∗ and Hk −X∗ ineach iteration of D-SDA are shown in Figure 1 (a), (b) and (c), respectively.

(v) For Example 5.5, the Frobenius norms of Ak, Gk and Hk+1 −Hk in eachiteration of C-SDA are shown in Figure 2 (a), (b) and (c), respectively. InFigure 2, the convergence behavior of Ak, Gk and Hk coincides with theresults of Theorem 4.2 (a). It shows that ‖A9‖F ≈ 20.3, ‖G9‖F ≈ 4.82×103

and ‖H9 −H8‖F ≈ 5.04 × 10−10. Furthermore, according to Figure 3 (a)and (b), HkAk converges to zero matrix as k →∞ and the matrix I+GkHk

is well-conditioned in each iteration so that Hk converges to the solution ofCARE. At the end of the algorithm, we have ‖H9A9‖F ≈ 5.04× 10−10 andcond(I +H9G9) ≈ 11.5 with ‖I +H9G9‖F ≈ 2.

(vi) In our numerical experience, SDAs do not break down. That is, the matrixI+GkHk is nonsingular in each iteration of SDAs. In fact, when the matrixI + GkHk is close to be ill-conditioned, the sequence Hk has convergedwell to the solutions of AREs.


0 10 2010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

k

|| A

k ||

F

(a)

0 10 2010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

k

|| G

k −

Y * ||

F

(b)

0 10 2010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

k

|| H

k −

X * ||

F

(c)

Figure 1. The Frobenius norms of Ak, Gk − Y∗ and Hk −X∗ forExample 5.1 by D-SDA.

6. Concluding remarks

In this paper, we propose the structured doubling algorithms for finding weakHermitian solutions of DAREs and CAREs. Under assumptions (A1) and (A2),we prove the global and linear convergence for SDA algorithms. The advantageof structured doubling algorithms is evident that the Hermitian solutions are ob-tained by the iterative process without any deflation preprocessing of unimodulareigenvalues. The MATLAB commands “care” and “dare” fail for the selected testexamples, because the initial matrices G0 and H0 are not both symmetric positivesemidefinite. Nevertheless the normalized residuals of Hermitian solutions of almostall tested examples computed by SDA algorithms are accurate to the machine ac-curacy. Numerical experiments show that SDA algorithms converge to the desiredHermitian solutions efficiently and reliably.

Appendix: Proof of Theorem 1.2

With α defined in (1.9), we have (I + GA−Hα HA−1

α )−1 being positive definite,and therefore (I +GA−H

α HA−1α )−1G is symmetric and positive semidefinite. This

implies that M − αL, where M, L are given in (1.5), is invertible and can befactorized as

M− αL =

[−αAα −αG−H AH

α

]=

[I −αGA−H

α

0 I

] [−αKα 0−H AH

α

](A.1)

with Kα ≡ Aα +GA−Hα H.


0 5 102

4

6

8

10

12

14

16

18

20

22

k

|| A

k ||

F

(a)

0 5 1010

−1

100

101

102

103

104

k

|| G

k ||

F

(b)

0 5 1010

−10

10−8

10−6

10−4

10−2

100

k

|| H

k+

1 − H

k ||

F

(c)

Figure 2. The Frobenius norms of Ak, Gk and Hk+1 − Hk forExample 5.5 by C-SDA.

Applying Cayley transformation to the symplectic pair (M,L) with parameterα, we get the Hamiltonian matrix

H = (M− αL)−1(M+ αL)

= −I + 2(M− αL)−1M≡[

A −G−H −AH

].(A.2)

With

G = 2A−1α

(I +GA−H

α HA−1α

)−1GA−H

α(A.3)

and

A = −(αA+ I + GH

)A−1

α .(A.4)

Let G = BBH ≥ 0 be a full rank decomposition. Since the Lagrangian subspaceis invariant via Cayley transformation, by applying the result of Theorem 1.1, it

suffices to show that (A, B) is c-stable. Let w 6= 0 ∈ Cn satisfy

wHB = 0, wHA = µwH .(A.5)

We shall prove that Re(µ) < 0.


0 5 1010

−10

10−8

10−6

10−4

10−2

100

k

|| H

k A

k ||

F

(a)

0 5 100

2

4

6

8

10

12

kco

nditi

on n

umbe

r of

I +

H k

G k

(b)

Figure 3. The Frobenius norms of HkAk and condition numbersof I +HkGk for Example 5.5 by C-SDA.

From (A.5), we have

wHA = −wH(αA+ I + GH

)A−1

α

= −wH(αA+ I)(I − αA)−1

= µwH .(A.6)

This implies

wHA = −α(

1 + µ

1− µ

)wH ,(A.7)

where wH = wH(I − αA)−1. From (A.3) and (A.5), it holds

wHG = wHA−1α G

(I +A−H

α HA−1α G

)−1A−H

α

= wHG(I +A−H

α HA−1α G

)−1A−H

α

= 0.(A.8)

From (A.8), it follows that

wHG = wHB = 0.(A.9)

In view of the d-stabilizability of the pair (A,B), we have∣∣∣∣1 + µ

1− µ

∣∣∣∣ < 1(A.10)

by (A.7). Therefore, Re(µ) < 0.


Acknowledgements

The authors thank Prof. E. K.-W. Chu from Monash University for many valu-able comments and helpful suggestions.

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Department of Mathematics, National Taiwan Normal University, Taipei 11677, Tai-

wan.

E-mail address: [email protected]

Department of Mathematics, National Tsing Hua University, Hsinchu 30043, Taiwan.

E-mail address: [email protected]

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