# An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

Authors: Feng Yin; Guang-Xin Huang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations(AXB-CYD,EXF-GYH)=(M,N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matrices X and Y. When the matrix equations are consistent, for any initial generalized reflexive matrix pair [X1,Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair [X̂,Ŷ] to a given matrix pair [X0,Y0] in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair [X̃*,Ỹ*] of a new corresponding generalized coupled Sylvester matrix equation pair (AX̃B-CỸD,EX̃F-GỸH)=(M̃,Ñ), where M̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.


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## Body

## 1. Introduction

In this paper, the following notations are used. Letℛm×n denote the set of all m×n real matrices. We denote by the superscript T the transpose of a matrix. In matrix space ℛm×n, define inner product as 〈A,B〉=tr(BTA) for all A,B∈ℛm×n, where tr(A) denotes the trace of a matrix A. ∥A∥ represents the Frobenius norm of A. ℛ(A) represents the column space of A. vec(·) represents the vector operator, that is, vec(A)=(a1T,a2T,…,anT)T∈ℛmn for the matrix A=(a1,a2,…,an)∈ℛm×n,ai∈Rm,i=1,2,…,n. A⊗B stands for the Kronecker product of matrices A and B, diag(A,B) denotes the block diagonal matrix with A and B and being the main diagonal elements orderly. In denotes the n-order identity matrix.Definition 1.1 (see [1, 2]).
A matrixP∈ℛn×n is said to be a generalized reflection matrix if P satisfies that PT=P,P2=I.Definition 1.2 (see [1, 2]).
LetP∈ℛn×n and Q∈ℛn×n be two generalized reflection matrices. A matrix A∈ℛn×n is called generalized reflexive (or generalized antireflexive) with respect to the matrix pair (P,Q) if PAQ=A(orPAQ=-A). The set of all n-by-n generalized reflexive matrices with respect to matrix pair (P,Q) is denoted by ℛrn×n(P,Q).The generalized reflexive and antireflexive matrices have many special properties and usefulness in engineering and scientific computations [1–6]. In particular, let P=Q, then a generalized reflexive matrix is called a reflexive matrix, which plays an important role in many areas and has been studied in [7–11]. Specially, let XT=X, then a reflexive matrix X is called a generalized bisymmetric matrix, which has been studied in [12, 13]. Moreover, let P=Q=Jn, then a generalized reflexive matrix is the well-known centrosymmetric matrix, which has been widely and extensively studied in [14–17].The generalized coupled Sylvester systems play a fundamental role in the various fields of engineering theory, particularly in control systems. The numerical solution of the generalized coupled Sylvester systems has been addressed in a large body of literature. Kågström and Westin [18] developed a generalized Schur method by applying the QZ algorithm to solve (AXB-CYD,EXF-GYH)=(M,N). Ding and Chen [19] presented an iterative least squares solutions of (AXB-CYD,EXF-GYH)=(M,N) based on a hierarchical identification principle [20], in addition, by applying the hierarchical identification principle,  Kılıçman and Zhour [21] developed an iterative algorithm for obtaining the weighted least-squares solution. Recently, some finite iterative algorithms have also been developed to solve matrix equations. For more detail, we refer to [11, 13, 22–30]. Wang [31, 32] gave the bi(skew)symmetric and centrosymmetric solutions to the system of quaternion matrix equations A1X=C1,A3XB3=C3. Wang [33] also solved a system of matrix equations over arbitrary regular rings with identity. Chang and Wang [34] gave the necessary and sufficient conditions for the existence of and the expressions for the symmetric solutions of the matrix equations AX+YA=C,AXAT+BYBT=C, and (ATXA,BTXB)=(C,D). Ding and Chen [25] also presented the gradient-based iterative algorithms by applying the gradient search principle and the hierarchical identification principle for the general coupled matrix equations ∑j=1pAijXjBij=Mi,i=1,2,…,p. Zhou et al. [35] proposed gradient-based iterative algorithms for solving the general coupled matrix equations with weighted least squares solutions. Wu et al. [36, 37] gave the finite iterative solutions to coupled Sylvester-conjugate matrix equations. Wu et al. [38] gave the finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns. Jonsson and Kågström [39] proposed recursive block algorithms for solving the one-sided and coupled Sylvester matrix equations (AX-YB,DX-YE)=(C,F). Jonsson and Kågström [40] also proposed recursive block algorithms for the two-sided and generalized Sylvester and Lyapunov matrix equations. Dehghan and Hajarian [7, 8] gave the reflexive and generalized bisymmetric matrices solutions of the generalized coupled Sylvester matrix equations (AY-ZB,CY-ZD)=(E,F). Very recently, Dehghan and Hajarian [12] constructed an iterative algorithm to solve the generalized coupled Sylvester matrix equations (AXB+CYD,EXF+GYH)=(M,N) over generalized bisymmetric matrices. Huang et al. [13] present an iterative algorithm for the generalized coupled Sylvester matrix equations (AY-ZB,CY-ZD)=(E,F) and its optimal approximation problem over generalized reflexive matrices solutions. In [30], the similar but different iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations (AXB-CYD,EXF-GYH)=(M,N) and the optimal approximation problem over reflexive matrices. However, the generalized coupled Sylvester matrix equations (AXB-CYD,EXF-GYH)=(M,N) and the optimal approximation over generalized reflexive matrices have not been solved.In this paper, we will consider the following two problems.Problem 1.
LetP∈ℛm×m,Q∈ℛn×n,R∈ℛs×s, and S∈ℛt×t be generalized reflection matrices. For given matrices A∈ℛp×m, B∈ℛn×q, C∈ℛp×s, D∈ℛt×q,M∈ℛp×q, E∈ℛk×m, F∈ℛn×l, G∈ℛk×s, H∈ℛt×l, N∈ℛk×l, find a pair of matrices X∈ℛrm×n(P,Q), Y∈ℛrs×t(R,S) such that
(1.1)AXB-CYD=M,EXF-GYH=N.Problem 2.
When Problem1 is consistent, let SE denote the set of the generalized reflexive solutions of Problem 1, that is,
(1.2)SE={[X,Y]∣AXB-CYD=M,EXF-GYH=N,Y∈Rrm×n(P,Q),Z∈Rrs×t(R,S)}.
For a given matrix pair [Y0,Z0]∈ℛrm×n(P,Q)×ℛrs×t(R,S), find [Ŷ,Ẑ]∈SE such that
(1.3)‖Ŷ-Y0‖2+‖Ẑ-Z0‖2=min⁡[Y,Z]∈SE{‖Y-Y0‖2+‖Z-Z0‖2}.The two-sided and generalized coupled Sylvester matrix equations (1.1) play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. In addition, as special type of generalized coupled Sylvester matrix equations (1.1), the generalized Sylvester matrix equation (AX-YB,CX-YD)=(E,F) arises in computing the deflating subspace of descriptor linear systems [18]. Wu et al. [36] presented some examples to show a motivation for studying (1.1).  Problem 2 occurs frequently in experiment design, see for instance [41].This paper is organized as follows. In Section2, we will solve Problem 1 by constructing an iterative algorithm, that is, if Problem 1 is consistent, then for an arbitrary initial matrix pair [Y1,Z1]∈ℛrm×n(P,Q)×ℛrs×t(R,S), we can obtain a solution pair [Y*,Z*] of Problem 1 within finite iterative steps in the absence of round-off errors. Let X1=ATKBT+ETLFT+PATKBTQ+PETLFTQ and Y1=-CTKDT-GTLHT-RCTKDTS-RGTLHTS, where K∈ℛp×q, L∈ℛk×l are arbitrary matrices, or more especially, let X1=0 and Y1=0, we can obtain the least Frobenius norm solutions of Problem 1. Then, in Section 3, we give the optimal approximate solution pair of Problem 2 by finding the least Frobenius norm generalized reflexive solution pair of the corresponding generalized coupled Sylvester matrix equations. In Section 4, several numerical examples are given to illustrate the application of our method. At last, some conclusions are drawn in Section 5.


## 2. An Iterative Algorithm for Solving Problem1

In this section, we will first introduce an iterative algorithm to solve Problem1, then prove that it is convergent. Then, we will give the least-norm generalized reflexive solutions of Problem 1 when an appropriate initial iterative matrix pair is chosen.For the purpose of simplification, we introduce the following operators:(2.1)Φ(X,Y)=AXB-CYD,Ψ(X,Y)=EXF-GYH.Algorithm 2.1.
We have the following steps.Step 1.
Input matricesA∈ℛp×m, B∈ℛn×q, C∈ℛp×s, D∈ℛt×q, M∈ℛp×q, E∈ℛk×m, F∈ℛn×l, G∈ℛk×s, H∈ℛt×l, N∈ℛk×l, and four generalized reflection matrices P∈ℛm×m, Q∈ℛn×n, R∈ℛs×s,S∈ℛt×t.Step 2.
Choose two arbitrary matricesX1∈ℛrm×n(P,Q), Y1∈ℛrs×t(R,S). Compute
(2.2)R1=diag⁡(M-Φ(X1,Y1),N-Ψ(X1,Y1)),U1=12[AT(M-Φ(X1,Y1))BT+ET(N-Ψ(X1,Y1))FT+PAT(M-Φ(X1,Y1))BTQ+PET(N-Ψ(X1,Y1))FTQ],V1=12[-CT(M-Φ(X1,Y1))DT-GT(N-Ψ(X1,Y1))HT-RCT(M-Φ(X1,Y1))DTS-RGT(N-Ψ(X1,Y1))HTS],k∶=1.Step 3.
IfRk=0, then stop and [Xk,Yk] is the solution of the generalized coupled Sylvester matrix equation (1.1); else if Rk≠0, but Uk=0 and Vk=0, then stop and the generalized coupled Sylvester matrix equations (1.1) are not consistent over generalized reflexive matrices; else k:=k+1.Step 4.
Compute(2.3)Xk=Xk-1+‖Rk-1‖2‖Uk-1‖2+‖Vk-1‖2Uk-1,Yk=Yk-1+‖Rk-1‖2‖Uk-1‖2+‖Vk-1‖2Vk-1,Rk=diag⁡(M-Φ(Xk,Yk),N-Ψ(Xk,Yk))=Rk-1-‖Rk-1‖2‖Uk-1‖2+‖Vk-1‖2diag⁡(Φ(Uk-1,Vk-1),Ψ(Uk-1,Vk-1)),Uk=12[AT(M-Φ(Xk,Yk))BT+ET(N-Ψ(Xk,Yk))FT+PAT(M-Φ(Xk,Yk))BTQ+PET(N-Ψ(Xk,Yk))FTQ]+‖Rk‖2‖Rk-1‖2Uk-1,Vk=12[-CT(M-Φ(Xk-1,Yk-1))DT-GT(N-Ψ(Xk-1,Yk-1))HT-RCT(M-Φ(Xk-1,Yk-1))DTS-RGT(N-Ψ(Xk-1,Yk-1))HTS]+‖Rk‖2‖Rk-1‖2VK-1.Step 5.
Go to Step3.
Obviously, it can be seen thatXk,Uk∈Rrm×n(P,Q), Yk,Vk∈Rrs×t(R,S), where k=1,2,….Lemma 2.2.
For the sequences{Ri},  {Ui}, and  {Vi} generated by Algorithm 2.1, and  s≥2, we have(2.4)tr⁡(RiTRj)=0,tr⁡(UiTUj+ViTVj)=0,i,j=1,2,…,s,i≠j.The proof of Lemma2.2 is presented in Appendix A.Lemma 2.3.
Suppose[X*,Y*] is an arbitrary solution pair of Problem 1, then for any initial generalized reflexive matrix pair  [X1,Y1], we have
(2.5)tr⁡((X*-Xi)TUi+(Y*-Yi)TVi)=‖Ri‖2,k=1,2,…,
where the sequences {Xi},{Yi}, {Ui}, {Vi}, and {Ri} are generated by Algorithm 2.1.The proof of Lemma2.3 is presented in Appendix B.Remark 2.4.
If there exist, a positive numberk such that Uk=0 and Vk=0 but Rk≠0, then by Lemma 2.3, we have that the generalized coupled Sylvester matrix equations (1.1) are not consistent over generalized reflexive matrices.Theorem 2.5.
Suppose that Problem1 is consistent, then for an arbitrary initial matrix pair  [X1,Y1]∈ℛrm×n(P,Q)×ℛrs×t(R,S), a generalized reflexive solution pair of Problem 1 can be obtained with finite iteration steps in the absence of round-off errors.Proof.
IfRi≠0,i=1,2,…,pq+st, by Lemma 2.3, we have Ui≠0,Vi≠0,i=1,2,…,pq+st, then we can compute [Xpq+st+1,Ypq+st+1] by Algorithm 2.1.
By Lemma2.2, we have(2.6)tr⁡(Rpq+st+1TRi)=0,i=1,2,…,pq+st,tr⁡(RiTRj)=0,i,j=1,2,…,pq+st,i≠j.
It can be seen that the set ofR1,R2,…,Rpq+st is an orthogonal basis of the matrix subspace(2.7)S={L∣L=diag⁡(L1,L2),L1∈Rp×q,L2∈Rs×t},
which implies that Rpq+st+1=0, that is, [Xpq+st+1,Ypq+st+1]∈ℛrm×n(P,Q)×ℛrs×t(R,S) is a solution pair of Problem 1. This completes the proof.To show the least Frobenius norm generalized reflexive solutions of Problem1, we first introduce the following result.Lemma 2.6 (see [42, Lemma 2.4]).
Suppose that the consistent system of linear equationAx=b has a solution  x*∈R(AT), then  x* is a unique least Frobenius norm solution of the system of linear equation.By Lemma2.6, the following result can be obtained.Theorem 2.7.
Suppose that Problem1 is consistent. If we choose the initial iterative matrices  X1=ATKBT+ETLFT+PATKBTQ+PETLFTQ and  Y1=-CTKDT-GTLHT-RCTKDTS −RGTLHTS, where  K∈ℛp×q,L∈ℛk×l are arbitrary matrices, especially,  X1=0∈ℛm×n(P,Q) and  Y1=0∈ℛs×t(R,S), then the solution pair  [Y*,Z*] generated by Algorithm 2.1 is the unique least Frobenius norm generalized reflexive solutions of Problem 1.Proof.
We know the solvability of the generalized coupled Sylvester matrix equations (1.1) over generalized reflexive matrices is equivalent to the following matrix equations:
(2.8)AXB-CYD=M,EXF-GYH=N,APXQB-CRYSD=M,EPXQF-GRYSH=N.
Then, the system of matrix equations (2.8) is equivalent to
(2.9)(BT⊗A-DT⊗CFT⊗E-HT⊗GBTQ⊗AP-DTS⊗CRFTQ⊗EP-HTS⊗GR)(vec⁡(X)vec⁡(Y))=(vec⁡(M)vec⁡(N)vec⁡(M)vec⁡(N)).
Let X1=ATKBT+ETLFT+PATKBTQ+PETLFTQ and Y1=-CTKDT-GTLHT − RCTKDTS-RGTLHTS, where K∈ℛp×q,L∈ℛk×l are arbitrary matrices, then
(2.10)(vec⁡(X1)vec⁡(Y1))=(vec⁡(ATKBT+ETLFT+PATKBTQ+PETLFTQ)vec⁡(-CTKDT-GTLHT-RCTKDTS-RGTLHTS))=(B⊗ATF⊗ETQB⊗PATQF⊗PET-D⊗CT-H⊗GT-SD⊗RCT-SH⊗RGT)(vec⁡(K)vec⁡(L)vec⁡(K)vec⁡(L))=(BT⊗A-DT⊗CFT⊗E-HT⊗GBTQ⊗AP-DTS⊗CRFTQ⊗EP-HTS⊗GR)T(vec⁡(K)vec⁡(G)vec⁡(K)vec⁡(G))∈R((BT⊗A-DT⊗CFT⊗E-HT⊗GBTQ⊗AP-DTS⊗CRFTQ⊗EP-HTS⊗GR)T).Furthermore, we can see that all Xk,Yk generated by Algorithm 2.1 satisfy
(2.11)(vec⁡(Xk)vec⁡(Yk))∈R((BT⊗A-DT⊗CFT⊗E-HT⊗GBTQ⊗AP-DTS⊗CRFTQ⊗EP-HTS⊗GR)T),by Lemma 2.6, we know that [X*,Y*] is the least Frobenius norm generalized reflexive solution pair of the system of linear equations (2.9). Since vector operator is isomorphic, [X*,Y*] is the unique least Frobenius norm generalized reflexive solution pair of the system of matrix equations (2.8), then [X*,Y*] is the unique least Frobenius norm generalized reflexive solution pair of Problem 1.


## 3. The Solution of Problem2

In this section, we will show that the optimal approximate solutions of Problem2 for a given generalized reflexive matrix pair can be derived by finding the least Frobenius norm generalized reflexive solutions of the corresponding generalized coupled Sylvester matrix equations.When Problem1 is consistent, the set of generalized reflexive solutions of Problem 1 denoted by SE is not empty. For a given matrix pair [X0,Y0]∈ℛrm×n(P,Q)×ℛrs×t(R,S), we have(3.1){AXB-CYD=MEXF-GYH=N⟺{A(X-X0)B-C(Y-Y0)D=M-AX0B+CY0DE(X-X0)F-G(Y-Y0)H=N-EX0F+GY0H
Set X̃=X-X0,Ỹ=Y-Y0,M̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H, then Problem 2 is equivalent to that of finding the least Frobenius norm generalized reflexive solutions pair [X̃*,Ỹ*] of the corresponding generalized coupled Sylvester matrix equations(3.2)AX̃B-CỸD=M̃,EX̃F-GỸH=Ñ.
By using Algorithm 2.1, let initial iteration matrix X̃1=ATKBT+ETLFT+PATKBTQ+PETLFTQ and Ỹ1=-CTKDT-GTLHT-RCTKDTS-RGTLHTS, or more especially, let X̃1=0∈ℛrm×n(P,Q) and Ỹ1=0∈ℛrs×t(R,S), then we can get the least Frobenius norm generalized reflexive solution pair [X̃*,Ỹ*] of (3.2). Thus, the generalized reflexive solution pair of the problem 2 can be represented as [X̂,Ŷ]=[X̃*+X0,Ỹ*+Y0].


## 4. Numerical Experiments

In this section, we will show several numerical examples to illustrate our results. All the tests are performed by MATLAB 7.8.Example 4.1.
Consider the generalized reflexive solutions of the generalized coupled Sylvester matrix equationsAXB-CYD=M,EXY-GYH=N, where
(4.1)A=(13-57-92046-10-296-83622-3-55-22-1-1184-6-9-9),B=(48-54-15-23392-6-27-81),C=(6-57-9246-119-123-81364-15-515-13-1129-6-9),D=(718-6-45-233-12081694-58-29),E=(145-171-23-2541342-36-81-548),F=(13-582-115-6251327-97-96-5121),G=(12-58-55-73249-6-37-1211),H=(248-547-15-236392-65-27-8114-3-26),M=(519117717011632-1031583-100238282180010293308-514839-4932458-75311322683-762-1164258858408),N=(-2426964-26532092603-65247-919291788-13311547-17992712-1684-659-27301756-765).Let
(4.2)P=(0000-10001000-10001000-10000),Q=(0010000-110000-100),R=(0001010000-101000),S=(000100000100-1001000001000)be generalized reflection matrices.
We will find the generalized reflexive solutions of the matrix equationsAXB-CYD=M,EXY-GYH=N by using Algorithm 2.1. It can be verified that the matrix equations are consistent over generalized reflexive matrices and the solutions are(4.3)X*=(-29253111-173-731113-1-2529),Y*=(1416-13497097-3-8-8383411416).
Because of the influence of the error of calculation, the residual Ri is usually unequal to zero in the process of the iteration, where i=1,2,…. For any chosen positive number ε; however, small enough, for example, ε=1.0000e-010, whenever ∥Rk∥<ε, stop the iteration, Xk and Yk are regarded to be generalized reflexive solutions of the matrix equations AXB-CYD=M,EXY-GYH=N. Choose an initially iterative matrix pair [X1,Y1]∈ℛr5×4(P,Q)×ℛr4×5(R,S), such as
(4.4)X1=(-12246-13278-783-261-2412),Y1=(34-13791091-31-83-137134).
By Algorithm 2.1, we have
(4.5)X30=(-2.00009.00002.00005.00003.00001.000011.0000-1.00007.00003.0000-7.00003.000011.00001.00003.0000-1.0000-2.00005.00002.00009.0000),Y30=(14.000016.0000-1.00003.00004.00009.00007.000009.00007.0000-3.0000-8.0000-8.00003.00008.00003.00004.00001.000014.000016.0000),‖R30‖=2.9703e-012<ε.So we obtain the generalized reflexive solutions of the matrix equations AXB-CYD=M,EXY-GYH=N. The relative error of the solutions and the residual are shown in Figure 1, where the relative error REk=(∥Xk-X*∥+∥Yk-Y*∥)/(∥X*∥+∥Y*∥) and the residual Rk=∥Rk∥.
Let(4.6)X1=(00000000000000000000),Y1=(00000000000000000000),
by Algorithm 2.1, we have
(4.7)X*=X30=(-2.00009.00002.00005.00003.00001.000011.0000-1.00007.00003.0000-7.00003.000011.00001.00003.0000-1.0000-2.00005.00002.00009.0000),Y*=Y30=(14.000016.0000-1.00003.00004.00009.00007.000009.00007.0000-3.0000-8.0000-8.00003.00008.00003.00004.00001.000014.000016.0000),‖R30‖=8.2565e-012<ε.The relative error of the solutions and the residual are shown in Figure 2.Figure 1
The relative error of the solutions and the residual for Example4.1 with X1≠0,Y1≠0.Figure 2
The relative error of the solutions and the residual for Example4.1 with X1=0,Y1=0.Example 4.2.
Consider the unique least-norm generalized reflexive solutions of the matrix equations in Example4.1. Let
(4.8)K=(10120-1011-101201-30121-10-2-1),L=(-11-10501-1321-1-203201-36),X1=ATKBT+CTLDT+PATKBTQ+PCTLDTQ,Y1=-ETKFT-GTLHT-RETKFTS-RGTLHTS.By using Algorithm 2.1, we have the least-norm generalized reflexive solutions of the matrix equations AXB-CYD=M,EXY-GYH=N as follows:
(4.9)X*=X30=(-2.00009.00002.00005.00003.00001.000011.0000-1.00007.00003.0000-7.00003.000011.00001.00003.0000-1.0000-2.00005.00002.00009.0000),Y*=Y30=(14.000016.0000-1.00003.00004.00009.00007.000009.00007.0000-3.0000-8.0000-8.00003.00008.00003.00004.00001.000014.000016.0000),‖R30‖=2.3986e-011e-012<ε.
The relative error of the solutions and the residual are shown in Figure3.Figure 3
The relative error of the solutions and the residual for Example4.2.Example 4.3.
LetSE denote the set of all generalized reflexive solutions of the matrix equations in Example 4.1. For a given generalized reflexive matrices
(4.10)X0=(3-1223-2001-3-1-30032-22-3-1),Y0=(24-220130135-22-5220224),we will find [X̂,Ŷ]∈SE, such that
(4.11)‖X̂-X0‖+‖Ŷ-Y0‖=min⁡[X,Y]∈SE‖X-X0‖+‖Y-Y0‖,that is, find the optimal approximate generalized reflexive solution pair to the matrix pair [X0,Y0] in SE in Frobenius norm.
LetX̃=X-X0,Ỹ=Y-Y0,M̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H, by the method mentioned in Section 3, we can obtain the least-norm generalized reflexive solution pair [X̃*,Ỹ*] of the matrix equations AX̃B+CỸD=M̃,EX̃F+GỸH=Ñ by choosing the initial iteration matrices X̃1=0 and Ỹ1=0, then by Algorithm 2.1, we have that(4.12)X̃*=X̃30*=(-5.000010.00000.00003.0000-0.00003.000011.0000-1.00006.00006.0000-6.00006.000011.00001.0000-0.0000-3.0000-0.00003.00005.000010.0000),Ỹ*=Ỹ30*=(12.000012.00001.00001.00004.00008.00004.000008.00004.0000-8.0000-6.0000-10.00008.00006.00001.00004.0000-1.000012.000012.0000),‖R30‖=6.3482e-010<ε=1.0000e-010and the optimal approximate generalized reflexive solutions to the matrix pair [X0,Y0] in Frobenius norm are
(4.13)X̂=X̃30*+X0=(-2.00009.00002.00005.00003.00001.000011.0000-1.00007.00003.0000-7.00003.000011.00001.00003.0000-1.0000-2.00005.00002.00009.0000),Ŷ=Ỹ30*+Y0=(14.000016.0000-1.00003.00004.00009.00007.000009.00007.0000-3.0000-8.0000-8.00003.00008.00003.00004.00001.000014.000016.0000).The relative error of the solutions and the residual are shown in Figure 4, where the relative error REk=(∥X̃k+X0-X*∥+∥Ỹk+Y0-Y*∥)/(∥X*∥+∥Y*∥) and the residual Rk=∥Rk∥.Figure 4
The relative error of the solutions and the residual for Example4.3.


## 5. Conclusions

In this paper, an efficient iterative algorithm is presented to solve the generalized coupled Sylvester matrix equationsAXB-CYD=M,EXY-GYH=N over generalized reflexive matrix pair [X,Y]∈ℛrm×n(P,Q)×ℛrs×t(R,S). When the matrix equations AXB-CYD=M,EXY-GYH=N are consistent over generalized reflexive matrices X and Y, for any generalized reflexive initial iterative matrix pair [X1,Y1]∈ℛrm×n(P,Q)×ℛrs×t(R,S), the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors. Let initial matrices X1=ATKBT+ETLFT+PATKBTQ+PETLFTQ and Y1=-CTKDT-GTLHT-RCTKDTS-RGTLHTS, where K∈ℛp×q,L∈ℛk×l are arbitrary matrices, especially, let X1=0∈ℛrm×n(P,Q) and Y1=0∈ℛrs×t(R,S), the unique least-norm generalized reflexive solutions of the matrix equations can be derived. Furthermore, the optimal approximate solutions of AXB-CYD=M,EXY-GYH=N for a given generalized reflexive matrix pair [X0,Y0]∈ℛrm×n(P,Q)×ℛrs×t(R,S) can be derived by finding the least-norm generalized reflexive solutions of two new corresponding generalized coupled Sylvester matrix equations. Finally, several numerical examples are given to illustrate that our iterative algorithm is quite effective.The results presented in this paper generalize some previous results [7, 12, 13, 30]. When B=I,C=I,F=I,G=I, P=Q, and R=S, then our results reduce to those in [7]. When P=Q,R=S,XT=X, and YT=Y, the results in this paper reduce to those in [12]. When B=I,C=I,F=I, and G=I, then the results in this paper reduce to those in [13]. When P=Q and R=S, then the results in this paper reduce to those in [30].



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*Source: 152805-2012-08-26.xml*