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

Journal of Applied Mathematics Volume 2012, Article ID 152805,28pages doi:10.1155/2012/152805

Research Article

An Iterative Algorithm for the Generalized

Reflexive Solutions of the Generalized Coupled

Sylvester Matrix Equations

Feng Yin

1

_{and Guang-Xin Huang}

2

1_{School of Science, Sichuan University of Science and Engineering, Zigong 643000, China} 2_{College of Management Science, Geomathematics Key Laboratory of Sichuan Province,}

Chengdu University of Technology, Chengdu 610059, China

Correspondence should be addressed to Feng Yin,[email protected]

Received 14 November 2011; Revised 13 March 2012; Accepted 25 March 2012 Academic Editor: Alain Miranville

Copyrightq2012 F. Yin and G.-X. Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 matricesX andY. 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-offerrors, 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 pairX, Y to a given matrix pairX0, Y0 in Frobenius norm can be derived

by finding the least-norm generalized reflexive solution pairX∗,Y∗ of a new corresponding generalized coupled Sylvester matrix equation pairAXB −CYD, E XF −GYH M, N, where

MM−AX0BCY0D,NN−EX0FGY0H. Several numerical examples are given to show

the effectiveness of the presented iterative algorithm.

1. Introduction

In this paper, the following notations are used. Let Rm×n denote the set of all m×n real matrices. We denote by the superscriptT the transpose of a matrix. In matrix space Rm×n, define inner product asA, BtrBTAfor allA, B∈ Rm×n, where trAdenotes the trace of a matrixA.Arepresents the Frobenius norm ofA.RArepresents the column space ofA. vec·represents the vector operator, that is, vecA aT₁,a₂T, . . . ,aT_nT ∈ Rmn for the matrix

of matricesAandB, diagA, Bdenotes the block diagonal matrix withAandBand being the main diagonal elements orderly.I_ndenotes the n-order identity matrix.

Definition 1.1see1,2. A matrixP ∈ Rn×nis said to be a generalized reflection matrix ifP

satisfies thatPTP, P2_I.

Definition 1.2see1,2. LetP∈ Rn×nandQ∈ Rn×nbe two generalized reflection matrices. A

matrixA∈ Rn×nis called generalized reflexiveor generalized antireflexivewith respect to the matrix pairP, QifPAQAorPAQ−A. The set of all n-by-n generalized reflexive matrices with respect to matrix pairP, Qis denoted byRn_r×nP, Q.

The generalized reflexive and antireflexive matrices have many special properties and usefulness in engineering and scientific computations 1–6. In particular, letP Q, then a generalized reflexive matrix is called a reflexive matrix, which plays an important role in many areas and has been studied in7–11. Specially, letXT X, then a reflexive matrixX is called a generalized bisymmetric matrix, which has been studied in12,13. Moreover, let

P Q Jn, then a generalized reflexive matrix is the well-known centrosymmetric matrix,

which has been widely and extensively studied in14–17.

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. K˚agstr ¨om and Westin 18 developed a generalized Schur method by applying the QZ algorithm to solveAXB−CYD, EXF−GYH M, N. Ding and Chen19 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, Kılıc¸man 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 to11,13,22–30. Wang 31, 32 gave the biskewsymmetric and centrosymmetric solutions to the system of quaternion matrix equationsA1X C1, A3XB3 C3. Wang 33 also solved a system

of matrix equations over arbitrary regular rings with identity. Chang and Wang34gave 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

AT_{XA, B}T_{XB C, D. Ding and Chen25 also presented the gradient-based iterative}

algorithms by applying the gradient search principle and the hierarchical identification principle for the general coupled matrix equationsp_j1A_ijX_jB_ij M_i, 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.38gave the finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns. Jonsson and K˚agstr ¨om39proposed recursive block algorithms for solving the one-sided and coupled Sylvester matrix equations AX − YB, DX − YE C, F. Jonsson and K˚agstr ¨om40also proposed recursive block algorithms for the two-sided and generalized Sylvester and Lyapunov matrix equations. Dehghan and Hajarian7,8gave the reflexive and generalized bisymmetric matrices solutions of the generalized coupled Sylvester matrix equationsAY−ZB, CY−ZD E, F. Very recently, Dehghan and Hajarian

12 constructed an iterative algorithm to solve the generalized coupled Sylvester matrix equations AXBCYD, EXFGYH M, N over generalized bisymmetric matrices.

Huang et al.13present 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. In30, the similar but different iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equationsAXB−CYD, EXF−

GYH M, Nand the optimal approximation problem over reflexive matrices. However,

the generalized coupled Sylvester matrix equationsAXB−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, andS∈ Rt×tbe 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 ∈ R}k×s_{, H ∈ R}t×l_{, N ∈ R}k×l_{, find a pair of matricesX ∈ R}m×n

r P, Q,Y ∈ Rsr×tR, S

such that

AXB−CYDM,

EXF−GYHN. 1.1

Problem 2. When Problem 1is consistent, let SE denote the set of the generalized reflexive

solutions of Problem1, that is,

SEX, Y|AXB−CYDM, EXF−GYH N, Y ∈ Rmr×nP, Q, Z∈ Rsr×tR, S

. 1.2 For a given matrix pairY0, Z0∈ Rmr×nP, Q× Rrs×tR, S, findY, Z ∈SEsuch that

_Y−Y 0 2 Z−Z0 2 min Y,Z∈SE Y−Y0 2_Z−Z 02 . 1.3

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 equationAX −YB, CX−YD E, Farises in computing the deflating subspace of descriptor linear systems18. Wu et al.36presented some examples to show a motivation for studying1.1. Problem2 occurs frequently in experiment design, see for instance41.

This paper is organized as follows. In Section 2, we will solve Problem 1 by constructing an iterative algorithm, that is, if Problem1 is consistent, then for an arbitrary initial matrix pairY1, Z1 ∈ Rmr×nP, Q× Rsr×tR, S, we can obtain a solution pairY∗, Z∗

of Problem 1 within finite iterative steps in the absence of round-off errors. Let X1

AT_KBT_ET_LFT_PAT_KBT_QPET_LFT_QandY

1 −CTKDT−GTLHT−RCTKDTS−RGTLHTS,

whereK ∈ Rp×q,L ∈ Rk×l are arbitrary matrices, or more especially, letX1 0 and Y1 0,

we can obtain the least Frobenius norm solutions of Problem1. Then, in Section3, 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 Section4, several numerical examples are given to illustrate the application of our method. At last, some conclusions are drawn in Section5.

2. An Iterative Algorithm for Solving Problem

1

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 Problem1when an appropriate initial iterative matrix pair is chosen.

For the purpose of simplification, we introduce the following operators:

ΦX, Y AXB−CYD,

ΨX, Y EXF−GYH. 2.1

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 ∈ R}k×s_{, H ∈ R}t×l_{, N∈ R}k×l_{, and four generalized reflection matricesP ∈ R}m×m_{, Q∈}

Rn×n_{, R∈ R}s×s_{, S∈ R}t×t_.

Step 2. Choose two arbitrary matricesX1∈ Rmr×nP, Q,Y1∈ Rsr×tR, S. Compute

R1diagM−ΦX1, Y1, N−ΨX1, Y1, U1 1 2 AT_M−ΦX 1, Y1BTETN−ΨX1, Y1FT PAT_M−ΦX 1, Y1BTQPETN−ΨX1, Y1FTQ , V1 1 2 −CT_M−ΦX 1, Y1DT−GTN−ΨX1, Y1HT −RCT_M−ΦX 1, Y1DTS−RGTN−ΨX1, Y1HTS , k:1. 2.2

Step 3. IfR_k 0, then stop andX_k, Y_kis the solution of the generalized coupled Sylvester

matrix equation1.1; else ifRk/0, butUk 0 andVk 0, then stop and the generalized

coupled Sylvester matrix equations 1.1 are not consistent over generalized reflexive matrices; elsek:k1. Step 4. Compute XkXk−1 Rk−1 2 Uk−12Vk−12 Uk−1, YkYk−1 Rk−1 2 Uk−12Vk−12 Vk−1, RkdiagM−ΦXk, Yk, N−ΨXk, Yk Rk−1− Rk−1 2 Uk−12Vk−12

Uk 1₂ AT_M−ΦX k, YkBTETN−ΨXk, YkFT PAT_M−ΦX k, YkBTQPETN−ΨXk, YkFTQ Rk2 Rk−12 Uk−1, Vk 1₂ −CT_M−ΦX k−1, Yk−1DT−GTN−ΨXk−1, Yk−1HT −RCT_M−ΦX k−1, Yk−1DTS−RGTN−ΨXk−1, Yk−1HTS Rk2 Rk−12 VK−1. 2.3 Step 5. Go to Step3.

Obviously, it can be seen thatXk, Uk ∈ Rmr×nP, Q, Yk, Vk ∈ Rsr×tR, S, where k

1,2, . . ..

Lemma 2.2. For the sequences{Ri},{Ui}, and{Vi}generated by Algorithm2.1, ands≥2, we have

trRT_iR_j0, trUT_iU_jV_iTV_j0, i, j1,2, . . . , s, i /j. 2.4

The proof of Lemma2.2is presented in AppendixA.

Lemma 2.3. Suppose X∗_{, Y}∗ _{is an arbitrary solution pair of Problem 1, then for any initial}

generalized reflexive matrix pairX1, Y1, we have

trX∗−XiTUi Y∗−YiTVi

Ri2, k1,2, . . . , 2.5

where the sequences{X_i},{Y_i},{U_i},{V_i}, and{R_i}are generated by Algorithm2.1.

The proof of Lemma2.3is presented in AppendixB.

Remark 2.4. If there exist, a positive numberk such thatUk 0 andVk 0 butRk/0, then

by Lemma2.3, we have that the generalized coupled Sylvester matrix equations1.1are not consistent over generalized reflexive matrices.

Theorem 2.5. Suppose that Problem 1 is consistent, then for an arbitrary initial matrix pair

X1, Y1∈ Rmr×nP, Q× Rsr×tR, S, a generalized reflexive solution pair of Problem1can be obtained

with finite iteration steps in the absence of round-offerrors.

Proof. IfR_i/0, i 1,2, . . . , pqst, by Lemma2.3, we haveU_i/0, V_i/0, i 1,2, . . . , pqst,

then we can computeXpqst1, Ypqst1by Algorithm2.1.

By Lemma2.2, we have

trRT_pqst1R_i0, i1,2, . . . , pqst,

trRT_iR_j0, i, j1,2, . . . , pqst, i /j.

It can be seen that the set of R1, R2, . . . , Rpqst is an orthogonal basis of the matrix

subspace

SL|LdiagL1, L2, L1∈ Rp×q, L2∈ Rs×t

, 2.7

which implies that R_pqst1 0, that is, X_pqst1, Y_pqst1 ∈ Rm_r×nP, Q× Rs_r×tR, S is a solution pair of Problem1. This completes the proof.

To show the least Frobenius norm generalized reflexive solutions of Problem1, we first introduce the following result.

Lemma 2.6see42, Lemma 2.4. Suppose that the consistent system of linear equationAx b

has a solutionx∗ ∈RAT, thenx∗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 Problem1is consistent. If we choose the initial iterative matricesX1

AT_KBT_ET_LFT_PAT_KBT_QPET_LFT_QandY

1−CTKDT−GTLHT−RCTKDTS−RGTLHTS,

whereK ∈ Rp×q, L ∈ Rk×l are arbitrary matrices, especially,X1 0 ∈ Rm×nP, QandY1 0 ∈

Rs×t_{R, S, then the solution pairY}∗_{, Z}∗_{generated by Algorithm2.1is the unique least Frobenius}

norm generalized reflexive solutions of Problem1.

Proof. We know the solvability of the generalized coupled Sylvester matrix equations1.1

over generalized reflexive matrices is equivalent to the following matrix equations:

AXB−CYDM,

EXF−GYHN,

APXQB−CRYSDM,

EPXQF−GRYSHN.

2.8

Then, the system of matrix equations2.8is equivalent to

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ BT_{⊗A −D}T_⊗C FT_{⊗E −H}T_⊗G BT_{Q⊗AP −D}T_S⊗CR FT_{Q⊗EP −H}T_S⊗GR ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ vecX vecY ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ vecM vecN vecM vecN ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠. 2.9

LetX1ATKBTETLFTPATKBTQPETLFTQandY1−CTKDT−GTLHT−RCTKDTS−

RGT_LHT_{S, whereK∈ R}p×q_{, L∈ R}k×l_{are arbitrary matrices, then}

vecX1 vecY1 vecATKBTETLFTPATKBTQPETLFTQ vec−CTKDT−GTLHT−RCTKDTS−RGTLHTS B⊗AT _F⊗ET _QB⊗PAT _QF⊗PET −D⊗CT _−H⊗GT _−SD⊗RCT _−SH⊗RGT ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ vecK vecL vecK vecL ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ BT_{⊗A −D}T_⊗C FT_{⊗E −H}T_⊗G BT_{Q⊗AP −D}T_S⊗CR FT_{Q⊗EP −H}T_S⊗GR ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ T⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ vecK vecG vecK vecG ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ∈ R ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ BT_{⊗A −D}T_⊗C FT_{⊗E −H}T_⊗G BT_{Q⊗AP −D}T_S⊗CR FT_{Q⊗EP −H}T_S⊗GR ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ T⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . 2.10

Furthermore, we can see that allXk, Ykgenerated by Algorithm2.1satisfy

vecXk vecYk ∈ R ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ BT_{⊗A −D}T_⊗C FT_{⊗E −H}T_⊗G BT_{Q⊗AP −D}T_S⊗CR FT_{Q⊗EP −H}T_S⊗GR ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ T⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , 2.11

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 Problem1.

3. The Solution of Problem

2

In this section, we will show that the optimal approximate solutions of Problem 2 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 Problem1is consistent, the set of generalized reflexive solutions of Problem1

denoted byS_Eis not empty. For a given matrix pairX0, Y0 ∈ Rrm×nP, Q× Rsr×tR, S, we

have ⎧ ⎨ ⎩ AXB−CYDM EXF−GYHN ⇐⇒ ⎧ ⎨ ⎩ AX−X0B−CY −Y0DM−AX0BCY0D EX−X0F−GY−Y0HN−EX0FGY0H 3.1

SetX X−X0,YY−Y0,MM−AX0BCY0D,NN−EX0FGY0H, then Problem2

is equivalent to that of finding the least Frobenius norm generalized reflexive solutions pair

X∗_,Y∗_{of the corresponding generalized coupled Sylvester matrix equations}

AXB −CYD M,

EXF −GYH N. 3.2

By using Algorithm 2.1, let initial iteration matrix X1 ATKBT ETLFT PATKBTQ

PET_LFT_{Q and Y}

1 −CTKDT −GTLHT −RCTKDTS −RGTLHTS, or more especially, let

X1 0 ∈ Rmr×nP, Q and Y1 0 ∈ Rsr×tR, S, then we can get the least Frobenius norm

generalized reflexive solution pairX∗,Y∗of3.2. Thus, the generalized reflexive solution pair of the problem2can be represented asX, Y X∗X0,Y∗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−CYDM, EXY−GYHN, where

A ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 3 −5 7 −9 2 0 4 6 −1 0 −2 9 6 −8 3 6 2 2 −3 −5 5 −22 −1 −11 8 4 −6 −9 −9 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , B ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 4 8 −5 4 −1 5 −2 3 3 9 2 −6 −2 7 −8 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, C ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 6 −5 7 −9 2 4 6 −11 9 −12 3 −8 13 6 4 −15 −5 15 −13 −11 2 9 −6 −9 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , D ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 7 1 8 −6 −4 5 −2 3 3 −12 0 8 1 6 9 4 −5 8 −2 9 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ,

E ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 14 5 −1 7 1 −2 3 −2 5 4 13 4 2 −3 6 −8 1 −5 4 8 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, F ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 3 −5 8 2 −11 5 −6 2 5 13 2 7 −9 7 −9 6 −5 12 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, G ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 −5 8 −5 5 −7 3 2 4 9 −6 −3 7 −12 11 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, H ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2 4 8 −5 4 7 −1 5 −2 3 6 3 9 2 −6 5 −2 7 −8 1 1 4 −3 −2 6 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , M ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 519 1177 1701 1632 −103 1583 −100 2382 82 1800 1029 3308 −514 839 −493 2458 −753 1132 2683 −762 −1164 258 858 408 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , N ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −2426 964 −2653 2092 603 −65 247 −919 291 788 −1331 1547 −17 992 712 −1684 −659 −2730 1756 −765 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠. 4.1 Let P ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 0 −1 0 0 0 1 0 0 0 −1 0 0 0 1 0 0 0 −1 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Q ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 1 0 0 0 0 −1 1 0 0 0 0 −1 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, R ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 1 0 1 0 0 0 0 −1 0 1 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, S ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 1 0 0 0 0 0 1 0 0 −1 0 0 1 0 0 0 0 0 1 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 4.2

We will find the generalized reflexive solutions of the matrix equationsAXB−CYD

M, EXY−GYH Nby using Algorithm2.1. It can be verified that the matrix equations are

consistent over generalized reflexive matrices and the solutions are

X∗ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −2 9 2 5 3 1 11 −1 7 3 −7 3 11 1 3 −1 −2 5 2 9 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Y∗ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 14 16 −1 3 4 9 7 0 9 7 −3 −8 −8 3 8 3 4 1 14 16 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠. 4.3

Because of the influence of the error of calculation, the residualRiis usually unequal to zero

in the process of the iteration, wherei1,2, . . .. For any chosen positive numberε; 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−

CYDM, EXY−GYHN. Choose an initially iterative matrix pairX1, Y1∈ R5r×4P, Q×

R4×5 r R, S, such as X1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −1 2 2 4 6 −1 3 2 7 8 −7 8 3 −2 6 1 −2 4 1 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Y1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 3 4 −1 3 7 9 1 0 9 1 −3 1 −8 3 −1 3 7 1 3 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠. 4.4 By Algorithm2.1, we have X30 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −2.0000 9.0000 2.0000 5.0000 3.0000 1.0000 11.0000 −1.0000 7.0000 3.0000 −7.0000 3.0000 11.0000 1.0000 3.0000 −1.0000 −2.0000 5.0000 2.0000 9.0000 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Y30 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 14.0000 16.0000 −1.0000 3.0000 4.0000 9.0000 7.0000 0 9.0000 7.0000 −3.0000 −8.0000 −8.0000 3.0000 8.0000 3.0000 4.0000 1.0000 14.0000 16.0000 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, R302.9703e−012< ε. 4.5

So we obtain the generalized reflexive solutions of the matrix equations AXB−CYD

Figure1, where the relative error REk Xk−X∗Yk−Y∗/X∗Y∗ and the residualRkR_k. Let X1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Y1 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, 4.6 by Algorithm2.1, we have X∗_X 30 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −2.0000 9.0000 2.0000 5.0000 3.0000 1.0000 11.0000 −1.0000 7.0000 3.0000 −7.0000 3.0000 11.0000 1.0000 3.0000 −1.0000 −2.0000 5.0000 2.0000 9.0000 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Y∗_Y 30 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 14.0000 16.0000 −1.0000 3.0000 4.0000 9.0000 7.0000 0 9.0000 7.0000 −3.0000 −8.0000 −8.0000 3.0000 8.0000 3.0000 4.0000 1.0000 14.0000 16.0000 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, R308.2565e−012< ε. 4.7

The relative error of the solutions and the residual are shown in Figure2.

Example 4.2. Consider the unique least-norm generalized reflexive solutions of the matrix

equations in Example4.1. Let

K ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 1 2 0 −1 0 1 1 −1 0 1 2 0 1 −3 0 1 2 1 −1 0 −2 −1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , L ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −1 1 −1 0 5 0 1 −1 3 2 1 −1 −2 0 3 2 0 1 −3 6 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, X1ATKBTCTLDTPATKBTQPCTLDTQ, Y1−ETKFT−GTLHT−RETKFTS−RGTLHTS. 4.8

0 5 10 15 20 25 30 0 1 2 3 4 5 6 7 8 9 10 Relative error/residual k(iteration step) 10∗REk 0.001∗Rk

Figure 1: The relative error of the solutions and the residual for Example4.1withX1/0, Y1/0.

By using Algorithm2.1, we have the least-norm generalized reflexive solutions of the matrix equationsAXB−CYDM, EXY−GYHNas follows:

X∗_X 30 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −2.0000 9.0000 2.0000 5.0000 3.0000 1.0000 11.0000 −1.0000 7.0000 3.0000 −7.0000 3.0000 11.0000 1.0000 3.0000 −1.0000 −2.0000 5.0000 2.0000 9.0000 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Y∗_Y 30 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 14.0000 16.0000 −1.0000 3.0000 4.0000 9.0000 7.0000 0 9.0000 7.0000 −3.0000 −8.0000 −8.0000 3.0000 8.0000 3.0000 4.0000 1.0000 14.0000 16.0000 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , R302.3986e−011e−012< ε. 4.9

0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 Relative error/residual k(iteration step) 10∗REk 0.001∗Rk

Figure 2: The relative error of the solutions and the residual for Example4.1withX10, Y10.

Example 4.3. LetSEdenote the set of all generalized reflexive solutions of the matrix equations

in Example4.1. For a given generalized reflexive matrices

X0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 3 −1 2 2 3 −2 0 0 1 −3 −1 −3 0 0 3 2 −2 2 −3 −1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Y0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2 4 −2 2 0 1 3 0 1 3 5 −2 2 −5 2 2 0 2 2 4 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, 4.10

we will findX, Y∈SE, such that

_X−X

0Y−Y0 min

X,Y∈SEX−X0Y−Y0,

4.11

that is, find the optimal approximate generalized reflexive solution pair to the matrix pair

X0, Y0inSEin Frobenius norm.

LetX X−X0,Y Y−Y0,MM−AX0BCY0D,NN−EX0FGY0H, by the

pairX∗,Y∗of the matrix equationsAXB CYD M, E XF GYH Nby choosing the initial iteration matricesX10 andY10, then by Algorithm2.1, we have that

X∗_X∗ 30 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −5.0000 10.0000 0.0000 3.0000 −0.0000 3.0000 11.0000 −1.0000 6.0000 6.0000 −6.0000 6.0000 11.0000 1.0000 −0.0000 −3.0000 −0.0000 3.0000 5.0000 10.0000 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Y∗_Y∗ 30 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 12.0000 12.0000 1.0000 1.0000 4.0000 8.0000 4.0000 0 8.0000 4.0000 −8.0000 −6.0000 −10.0000 8.0000 6.0000 1.0000 4.0000 −1.0000 12.0000 12.0000 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠, R306.3482e−010< ε1.0000e−010 4.12

and the optimal approximate generalized reflexive solutions to the matrix pairX0, Y0 in

Frobenius norm are

XX∗ 30X0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −2.0000 9.0000 2.0000 5.0000 3.0000 1.0000 11.0000 −1.0000 7.0000 3.0000 −7.0000 3.0000 11.0000 1.0000 3.0000 −1.0000 −2.0000 5.0000 2.0000 9.0000 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , Y Y∗ 30Y0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 14.0000 16.0000 −1.0000 3.0000 4.0000 9.0000 7.0000 0 9.0000 7.0000 −3.0000 −8.0000 −8.0000 3.0000 8.0000 3.0000 4.0000 1.0000 14.0000 16.0000 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠. 4.13

The relative error of the solutions and the residual are shown in Figure4, where the relative errorREk XkX0−X∗YkY0−Y∗/X∗Y∗and the residualRkRk.

5. Conclusions

In this paper, an efficient iterative algorithm is presented to solve the generalized coupled Sylvester matrix equationsAXB−CYD M, EXY −GYH Nover generalized reflexive matrix pair X, Y ∈ Rm_r×nP, Q× R_rs×tR, 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 ∈ R_rm×nP, Q × Rs_r×tR, S,

0 5 10 15 20 25 30 0 2 4 6 8 10 12 Relative error/residual k(iteration step) 0.1∗REk 0.00001∗Rk

Figure 3: The relative error of the solutions and the residual for Example4.2.

0 5 10 15 20 25 30 0 2 4 6 8 10 12 Relative error/residual k(iteration step) 10∗REk 0.001∗Rk

Figure 4: The relative error of the solutions and the residual for Example4.3.

iterative steps in the absence of round-offerrors. Let initial matricesX1ATKBTETLFT

PAT_KBT_{Q PE}T_LFT_{Q and Y}

1 −CTKDT − GTLHT − RCTKDTS − RGTLHTS, where

K ∈ Rp×q_{, L ∈ R}k×l _{are arbitrary matrices, especially, let X}

1 0 ∈ Rm_r×nP, Q and

equations can be derived. Furthermore, the optimal approximate solutions ofAXB−CYD

M, EXY −GYH N for a given generalized reflexive matrix pairX0, Y0 ∈ Rmr×nP, Q×

Rs×t

r R, Scan 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 results7, 12,13,30. WhenB I, C I, F I, GI,P Q, andR S, then our results reduce to those in7. WhenP Q, R S, XT X, andYT Y, the results in this paper reduce to those in12. WhenB I, C I, F I, andG I, then the results in this paper reduce to those in13. WhenP QandRS, then the results in this paper reduce to those in30.

Appendices

A. The Proof of Lemma

2.2

Since trRT_iRj trRT_jRi, trUT_iUj trUT_jUi, and trV_iTVj trV_jTVi for all i, j

1,2, . . . , s, we only need to prove that

trRT_iRj

0, trU_iTUjV_iTVj

0, 1≤j < i≤s. A.1

We prove the conclusion by induction, and two steps are required.

Step 1. We will show that

trRT_i1Ri

0, trUT_i1UiViT1Vi

0, i1,2, . . . , s−1. A.2

To prove this conclusion, we also use induction. Fori1, by Algorithm2.1, we have that trRT₂R1 tr ⎛ ⎝ R1− R1 2 U12V12

diagΦU1, V1,ΨU1, V1

T R1 ⎞ ⎠ R12− R1 2 U12V12

trdiagΦU1, V1,ΨU1, V1

T ×diagM−ΦX1, Y1, N−ΨX1, Y1 R12− R1 2 U12V12 ×trΦU1, V1TM−ΦX1, Y1 ΨU1, V1TN−ΨX1, Y1 R12− R1 2 U12V12 trUT₁ATM−ΦX1, Y1BTUT1ETN−ΨX1, Y1FT −VT 1CTM−ΦX1, Y1DT−V1TGTN−ΨX1, Y1HT

R12− R1 2 U12V12 ×tr UT 1 AT_M−ΦX 1, Y1BTETN−ΨX1, Y1FT 2 ATM−ΦX1, Y1BTETN−ΨX1, Y1FT 2 PATM−ΦX1, Y1BTQPETN−ΨX1, Y1FTQ 2 −PATM−ΦX1, Y1BTQPETN−ΨX1, Y1FTQ 2 VT 1 −CT_M−ΦX 1, Y1DT−GTN−ΨX1, Y1HT 2 −CTM−ΦX1, Y1DT−GTN−ΨX1, Y1HT 2 −RCTM−ΦX1, Y1DTS−RGTN−ΨX1, Y1HTS 2 −−RCTM−ΦX1, Y1DTS−RGTN−ΨX1, Y1HTS 2 R12− R1 2 U12V12 ×tr UT 1 AT_M−ΦX 1, Y1BTETN−ΨX1, Y1FT 2 PATM−ΦX1, Y1BTQPETN−ΨX1, Y1FTQ 2 VT 1 −CT_M−ΦX 1, Y1DT−GTN−ΨX1, Y1HT 2 −RCTM−ΦX1, Y1DTS−RGTN−ΨX1, Y1HTS 2 R12− R1 2 U12V12 trUT₁U1V1TV1 0, trUT₂U1 trV₂TV1 tr AT_M−ΦX 2, Y2BTETN−ΨX2, Y2FT 2 PATM−ΦX2, Y2BTQPETN−ΨX2, Y2FTQ 2 R22 R12 U1 _T U1 ⎞ ⎠

tr −CT_M−ΦX 2, Y2DT−GTN−ΨX2, Y2HT 2 −RCTM−ΦX2, Y2DTS−RGTN−ΨX2, Y2HTS 2 R22 R12 V1 T V1 ⎞ ⎠ tr ⎛ ⎝ AT_M−ΦX 2, Y2BTETN−ΨX2, Y2FT R2 2 R12 U1 T U1 ⎞ ⎠ tr ⎛ ⎝ −CT_M−ΦX 2, Y2DT−GTN−ΨX2, Y2HTR2 2 R12 V1 T V1 ⎞ ⎠ trUT₁ATM−ΦX2, Y2BTETN−ΨX2, Y2FT VT 1 −CT_M−ΦX 2, Y2DT−GTN−ΨX2, Y2HT R22 R12 U12V12 trM−ΦX2, Y2TAU1B N−ΨX2, Y2TEU1F−M−ΦX2, Y2TCV1D −N−ΨX2, Y2TGV1H R22 R12 U12V12 trdiagM−ΦX2, Y2T,N−ΨX2, Y2T

diagΦU1, V1,ΨU1, V1 R22 R12 U12V12 U12V12 R12 trRT₂R1−R2 R22 R12 U12V12 0. A.3

Assume thatA.2holds forik−1, that is, trRT_kRk−1 0,trUT_kUk−1V_kTVk−1 0.

Whenik, we have that

trRT_k1Rk tr ⎛ ⎝ Rk− Rk 2 Uk2Vk2

diagΦUk, Vk,ΨUk, Vk

T Rk ⎞ ⎠ Rk2− Rk 2 Uk2Vk2

trdiagΦU_k, V_k,ΨU_k, V_kT

Rk2− Rk 2 Uk2Vk2 ×trΦU_k, V_kTM−ΦX_k, Y_k ΨU_k, V_kTN−ΨX_k, Y_k Rk2− Rk 2 Uk2Vk2 trUT_kATM−ΦXk, YkBTUT_kETN−ΨXk, YkFT −VT kCTM−ΦXk, YkDT−VkTGTN−ΨXk, YkHT Rk2− Rk 2 Uk2Vk2 tr UT k AT_{M−ΦXk, YkB}T_ET_{N−ΨXk, YkF}T 2 ATM−ΦXk, YkBTETN−ΨXk, YkFT 2 PATM−ΦXk, YkBTQPETN−ΨXk, YkFTQ 2 −PATM−ΦXk, YkBTQPETN−ΨXk, YkFTQ 2 VT k −CT_M−ΦX k, YkDT−GTN−ΨXk, YkHT 2 −CTM−ΦXk, YkDT−GTN−ΨXk, YkHT 2 −RCTM−ΦXk, YkDTS−RGTN−ΨXk, YkHTS 2 −−RCTM−ΦXk,YkDTS−RGTN−ΨXk,YkHTS 2 Rk2− Rk 2 Uk2Vk2 tr UT k AT_M−ΦX k, YkBTETN−ΨXk, YkFT 2 PATM−ΦXk, YkBTQPETN−ΨXk, YkFTQ 2 VT k −CT_{M−ΦXk, YkD}T_−GT_{N−ΨXk, YkH}T 2 −RCTM−ΦXk, YkDTS−RGTN−ΨXk, YkHTS 2 Rk2− Rk 2 Uk2Vk2 trUT_kU_kV_kTV_k0, trUT_k1Uk trV_kT₁Vk

tr AT_M−ΦX k1, Yk1BTETN−ΨXk1, Yk1FT 2 PATM−ΦXk1, Yk1BTQPETN−ΨXk1, Yk1FTQ 2 Rk12 Rk2 Uk T Uk ⎞ ⎠ tr −CT_{M−ΦXk1, Yk1D}T_−GT_{N−ΨXk1, Yk1H}T 2 −RCTM−ΦXk1, Yk1DTS−RGTN−ΨXk1, Yk1HTS 2 Rk12 Rk2 Vk T Vk ⎞ ⎠ tr ⎛ ⎝ AT_M−ΦX k1, Yk1BTETN−ΨXk1, Yk1FTRk1 2 Rk2 Uk _T Uk ⎞ ⎠ tr ⎛ ⎝ −CT_M−ΦX k1, Yk1DT−GTN−ΨXk1, Yk1HTRk1 2 Rk2 Vk T Vk ⎞ ⎠ trUT_kATM−ΦX_k1, Y_k1BTETN−ΨX_k1, Y_k1FT VT k −CT_M−ΦX k1, Yk1DT−GTN−ΨXk1, Yk1HT Rk12 Rk2 Uk2Vk2 trM−ΦXk1, Yk1TAUkB N−ΨXk1, Yk1TEUkF −M−ΦXk1, Yk1TCVkD −N−ΨXk1, Yk1TGVkH Rk12 Rk2 Uk2Vk2 trdiagM−ΦXk1, Yk1T,N−ΨXk1, Yk1T

diagΦUk, Vk,ΨUk, Vk

Rk12 Rk2 Uk2Vk2 Uk2Vk2 Rk2 trRT_k1Rk−Rk1 Rk12 Rk2 Uk2Vk2 0. A.4 Hence,A.2holds forik. Therefore,A.2holds by the principle of induction.

Step 2. We show that

trRT_i1R_j0, trUT_i1U_jV_iT₁V_j0, j1,2, . . . , i, ∀i≥1. A.5

Wheni1,A.5holds. Assume that trRT_iRj 0, trU_iTUjViTVj 0, j 1,2, . . . , s−1, ∀s≥2, A.6

then we show that

trRT_i1Rj

0, trUT_i1UjViT1Vj

0, j 1,2, . . . , s. A.7

In fact, we have that

trRT_i1Rj tr ⎛ ⎝ Ri− Ri 2 Ui2Vi2

diagΦUi, Vi,ΨUi, Vi

T Rj ⎞ ⎠ trRT_iR_j− Ri 2 Ui2Vi2

trdiagΦU_i, V_i,ΨU_i, V_iT

×diagM−ΦX_j, Y_j, N−ΨX_j, Y_j − Ri2 Ui2Vi2 trΦU_i, V_iTM−ΦX_j, Y_j ΨU_i, V_iTN−ΨX_j, Y_j − Ri2 Ui2Vi2 trUT_iATM−ΦX_j, Y_jBTU_iTETN−ΨX_j, Y_jFT −VT i CT M−ΦXj, YjDT−ViTGT N−ΨXj, YjHT − Ri2 Ui2Vi2 tr UT i AT_M−ΦX j, YjBTETN−ΨXj, YjFT 2 AT M−ΦXj, YjBTETN−ΨXj, YjFT 2 PAT M−ΦXj, YjBTQPETN−ΨXj, YjFTQ 2 −PAT M−ΦXj, YjBTQPETN−ΨXj, YjFTQ 2

VT i −CT_{M−ΦXj, YjD}T_−GT_{N−ΨXj, YjH}T 2 −CT M−ΦXj, YjDT−GTN−ΨXj, YjHT 2 −RCT M−ΦXj, YjDTS−RGTN−ΨXj, YjHTS 2 −−RCT M−ΦXj, YjDTS−RGTN−ΨXj, YjHTS 2 − Ri2 Ui2Vi2 tr UT i AT_{M−ΦXj, YjB}T_ET_{N−ΨXj, YjF}T 2 PAT M−ΦXj, YjBTQPETN−ΨXj, YjFTQ 2 VT i −CT_{M−ΦXj, YjD}T_−GT_{N−ΨXj, YjH}T 2 −RCT M−ΦXj, YjDTS−RGTN−ΨXj, YjHTS 2 − Ri2 Ui2Vi2 tr UT i Uj− _Rj2 _Rj−12Uj−1 VT i Vj− _Rj2 _Rj−12Vj−1 − Ri2 Ui2Vi2 trUT_iU_jV_iTV_j Ri 2_R j2 Ui2Vi2Rj−14 ×trUT_iUj−1 trV_iTVj−1 0. A.8

From the above results, we have trRT_i1Rj1 0, j 1,2, . . . , s−1, and

trUT_i1U_jtrV_iT₁V_j tr ⎛ ⎝ ⎡ ⎣ATM−ΦXi1, Yi1BTETN−ΨXi1, Yi1FT 2 PATM−ΦXi1, Yi1BTQPETN−ΨXi1, Yi1FTQ 2 Ri12 Ri2 Ui _T Uj ⎞ ⎠

tr ⎛ ⎝ −CT_M−ΦX i1, Yi1DT−GTN−ΨXi1, Yi1HT 2 −RCTM−ΦXi1, Yi1DTS−RGTN−ΨXi1, Yi1HTS 2 Ri12 Ri2 Vi _T Vj ⎞ ⎠ tr ⎛ ⎝ AT_M−ΦX i1, Yi1BTETN−ΨXi1, Yi1FTRi1 2 Ri2 Ui _T Uj ⎞ ⎠ tr ⎛ ⎝ −CT_M−ΦX i1, Yi1DT−GTN−ΨXi1, Yi1HTRi1 2 Ri2 Vi _T Vj ⎞ ⎠ trUT_jATM−ΦXi1, Yi1BTETN−ΨXi1, Yi1FT VT j −CT_M−ΦX i1, Yi1DT−GTN−ΨXi1, Yi1HT Ri12 Ri2 trUT_iU_jtrV_iTV_j trM−ΦXi1, Yi1TAUjB N−ΨXi1, Yi1TEUjF −M−ΦXi1, Yi1TCVjD −N−ΨXi1, Yi1TGVjH Ri12 Ri2 trUT_iUj trV_iTVj trdiagM−ΦXi1, Yi1T,N−ΨXi1, Yi1T diagΦUj, Vj,ΨUj, Vj Ri12 Ri2 trUT_iUj trV_iTVj Uj 2_V j2 _Rj2 tr RT i1 Rj−Rj1 Ri1 2 Ri2 trUT_iUj trV_iTVj 0. A.9 By the principle of induction,A.5holds.

Noting that A.1 is implied in Steps 1 and 2 by the principle of induction. This completes the proof.

B. The Proof of Lemma

2.3

Fori1, we have that trX∗−X1TU1 Y∗−Y1TV1 tr X∗_−X 1T AT_M−ΦX 1, Y1BTETN−ΨX1, Y1FT 2 PATM−ΦX1, Y1BTQPETN−ΨX1, Y1FTQ 2 Y∗_−Y 1T −CT_M−ΦX 1, Y1DT−GTN−ΨX1, Y1HT 2 −RCTM−ΦX1, Y1DTS−RGTN−ΨX1, Y1HTS 2 trX∗−X1T AT_M−ΦX 1, Y1BTETN−ΨX1, Y1FT Y∗_−Y 1T −CT_M−ΦX 1, Y1DT−GTN−ΨX1, Y1HT trM−ΦX1, Y1TAX∗−X1B N−ΨX1, Y1TEX∗−X1F −M−ΦX1, Y1TCZ∗−Z1D−N−ΨX1, Y1TGY∗−Y1H tr M−ΦX1, Y1T 0 0 N−ΨX1, Y1T AX∗_−X1B−CY∗_{−Y1D 0} 0 EX∗−X1F−GY∗−Y1H tr ⎛ ⎝ M−ΦX1, Y1 0 0 N−ΨX1, Y1 T_M− ΦX1, Y1 0 0 N−ΨX1, Y1 ⎞ ⎠ R12. B.1

Assume that2.5holds forik. Whenik1, by Algorithm2.1, we have that trX∗−Xk1TUk1 Y∗−Yk1TVk1 tr X∗_−X k1T AT_{M−ΦXk1, Yk1B}T_ET_{N−ΨXk1, Yk1F}T 2 PATM−ΦXk1, Yk1BTQPETN−ΨXk1, Yk1FTQ 2 Rk12 Rk2 Uk

Y∗_−Y k1T −CT_M−ΦX k1, Yk1DT−GTN−ΨXk1, Yk1HT 2 −RCTM−ΦXk1, Yk1DTS−RGTN−ΨXk1, Yk1HTS 2 Rk12 Rk2 Vk trX∗−X_k1TATM−ΦX_k1, Y_k1BTETN−ΨX_k1, Y_k1FT Y∗_−Y k1T −CT_M−ΦX k1, Yk1DT−GTN−ΨXk1, Yk1HT Rk12 Rk2 trX∗−Xk1TUk Y∗−Yk1TVk trM−ΦX_k1, Y_k1TAX∗−X_k1B N−ΨX_k1, Y_k1TEX∗−X_k1F −M−ΦXk1, Yk1TCZ∗−Zk1D−N−ΨXk1, Yk1TGY∗−Yk1H Rk12 Rk2 trX∗−Xk1TUk Y∗−Yk1TVk tr M−ΦXk1, Yk1T 0 0 N−ΨX_k1, Y_k1T AX∗_−X k1B−CY∗−Yk1D 0 0 EX∗−Xk1F−GY∗−Yk1H Rk12 Rk2 trX∗−Xk1TUk Y∗−Yk1TVk tr M−ΦXk1, Yk1T 0 0 N−ΨXk1, Yk1T M−ΦXk1, Yk1 0 0 N−ΨX_k1, Y_k1 Rk12 Rk2 trX∗−X_k1TU_k Y∗−Y_k1TV_k Rk12Rk1 2 Rk2 trX∗−X_kTU_k Y∗−Y_kTV_k − Rk12 Uk2Vk2 trUT_kU_kV_kTV_kR_k12. B.2

Therefore, 2.5 holds fori k 1. Thus, 2.5 holds by the principal of induction. This completes the proof.

Acknowledgments

The authors are very much indebted to the anonymous referees and their editors for their constructive and valuable comments and suggestions which greatly improved the original copy of this paper. Grateful acknowledgements are given to Professor. Alain Miranville for his comments and suggestions that helped improve the second version of this paper greatly. This work was partially supported by the Research Fund ProjectNatural Science 2010XJKYL018and Natural Science Foundation of Sichuan Education Department

12ZB289respectively. This work is also supported by Open Fund of Geomathematics Key Laboratory of Sichuan Province scsxdz2011005 and Key Natural Science Foundation of Sichuan Education Department12ZA008.

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