World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:10, No:11, 2016

Conjugate Gradient Algorithm for the Symmetric Arrowhead Solution of Matrix Equation AXB=C Minghui Wang, Luping Xu, Juntao Zhang

 nn set is denoted as AR . If aaii11 (1,2,,)in  , then we Abstract—Based on the conjugate gradient (CG) algorithm, the denote that A is symmetric arrow-head matrix, this type of constrained matrix equation AXB=C and the associate optimal nn approximation problem are considered for the symmetric arrowhead matrix set is denoted as SAR . veci  A denotes a vector matrix solutions in the premise of consistency. The convergence T results of the method are presented. At last, a numerical example is aa11,,,,,,, 21 aaannn 1 22 33 a . given to illustrate the efficiency of this method. Symmetric arrowhead matrices have many applications in the modern control theory which can represent the parameter Keywords—Iterative method, symmetric arrowhead matrix, matrix of nonlinear control systems or the large sparse matrix in conjugate gradient algorithm. the linear systems [1], [5]-[7]. With the development of electromagnetic compatibility, the mathematical representation I. INTRODUCTION of the influence factors of electromagnetic interference also has ET Rmn be the set of mn real matrices, SARnn be the set potential application value. In this paper, we consider the

Lof nn real symmetric arrowhead matrices and In be the following constrained matrix equation T † identity matrix of order n . For any A Rmn , A , A , A and F AXB C (1) A denote the transpose, Moore-Penrose generalized inverse, 2 Frobenius norm and Euclid norm, respectively. in which A Rmn , B  Rns , C  R ms . The above matrix For any AB,  Rmn , AB,0 traceB T A denotes the equation and other constrained matrix equations have been mn studied in [2], [3], [8], [9], etc. Peng et al. [4] analyzed CG inner product of A and B . Therefore, R is a complete algorithm to obtain corresponding symmetric solutions, 2 inner product space endowed with A  AA, . For any skew-symmetric solutions, centro-symmetric solutions and so mn on. Based on the classical method, we will utilize the operable non-zero matrices AA12,,, Ak  R , if iterative method to find the symmetric arrowhead matrix A, A trace AT Aij0 , then it is easy to verify ji i j    solution of the matrix equation (1).

that AA12,,, Ak are linearly independent and orthogonal. II. THE CONJUGATE GRADIENT ALGORITHM Proposition 1. Let AB,  Rnn , then T In this section, by means of the study of the classical CG trace() A trace ( A );trace() AB trace () BA algorithm for solving the linear matrix equation in [4], we trace( A B ) trace () A  trace () B propose the following algorithm to solve the matrix equation (1) nn for the symmetric arrowhead solution and give some main Definition 1. If a matrix Aa ij  R satisfies the following results in detail. form: Firstly, we define the following linear operator:

aaa11 12 13 a 1n nn nn  RAR aa21 22 00  :  (2)   XX Aa 00 a  31 33  in which  XEXXEdiagXEXE 11  11   2. 11 11 International Science Index, Mathematical and Computational Sciences Vol:10, No:11, 2016 waset.org/Publication/10006599 aannn1 00 According to the properties of the inner product matrix, it is

easy to verify that then we denote that A is arrowhead matrix, this type of matrix

XY,, X Y  X , Y

This work is supported by the Natural Science Foundation of Shandong Prowince in China (ZR2013FL014, ZR2016AB07) and the Science and nn nn Technology Program of Shandong Universitues of China(J13LN73) in which X  R , Y  AR . Here we discuss the iterative M. Wang, L. Xu andJ. Zhang are with Department of Mathematics, algorithm of the matrix equation (1) as: Qingdao University of Science & Technology, Shandong 266061, China (e-mail: [email protected]; [email protected]; [email protected]).

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Algorithm CG-W. Lemma 2. For k  2 , the sequencesRi  , Qi  generalized by nn Step1. Initialization. For initial matrix X1 SAR , compute Algorithm CG-W satisfy

TT RCAXB11 , trace Rij R  0, trace Q ij Q   0, ij,1,2,,,  kij (4) TT TTT ARB11  ARB P1  , ： 2 Proof We shall prove this lemma by induction. nn nn First, notice that Pi SR , Qi SAR , by Lemma 1 and Q1 P 1  E 11 P 1  PE 1 11  diag  P 1 2. E 11 PE 1 11 Algorithm CG-W, we obtain Step2. Iteration. For k  1, 2,, compute R 2 TT1 trace R21 R trace  R 11 R2 trace Q 11 P 2 Q1 Rk XX Q, 22 kk1 2 k RR Q 2211T k  RtraceQQRtraceQQ11111122  0, QQ 11 Step3. Compute as well as T RCAXB , T  kk11 trace P21 Q T  T trace Q21 Q trace P 22 Q 1 Q 1 ARTTTT B AR B Q  kk11 1  Pk 1  , 2 T  trace P Q trace P Q  0. T 21  21 trace Pkk1 Q QPkk11  Q k. Q 2 k Suppose that (4) holds for ks2 and notice that T trace Qss Q 1   0 .According to Lemma 1, we have if Rk 1  0 or Rk 1  0 , Qk 1  0 , stop; otherwise continue to

step (2). 22 TTRRss2 By algorithm CG-W it is clear that: trace Rss1 R trace  R ss R22 trace Q ss P  R s trace Q s P s QQ ss nn nn nn 2 T 2 R trace Pss Q 1 Pi SR , Qi SAR , X i SAR , i  1, 2, . s  RtraceQQQssss  1 QQ22 ss1 The following will demonstrate that the algorithm CG-W is 2 T 2 R trace Pss Q 1 s T T RtraceQQtsss22  race Qss Q 1 terminated by a finite iterative step.  QQss1 Lemma 1. The sequences R and Q generalized by 2  i   i  R RtraceQQ2 s T  0, Algorithm CG-W satisfy sss2  Qs

R 2 trace RTT R trace R Ri trace Q P, i, j  1,2, (3) and ij1  ij2  ij T T Qi  trace Pst1 Q T  trace Qss11 Q trace P s2 Q s Q s Q  T T s  Proof: Since PPii , QQii , by Algorithm CG-W, we have   T T  trace Pss11 Q trace P ss Q  0. T 2  R T T i trace Rij11 R trace C AX i B R j  trace C A X i2 Q i B R j Q i T Thus, by Lemma 1, it is clear that trace Rsj1 R   0 when R 2 trace C AX BT R i BTT Q A R T T International Science Index, Mathematical and Computational Sciences Vol:10, No:11, 2016 waset.org/Publication/10006599  ij2 ij j  1 . And we notice that trace Rsj R   0 , trace Qsj Q   0 , Qi T 2 TT TT T R QA RB QA RB trace Qsj Q 1  0 for j  2,3, ,s  1 . By Lemma 1 and T i ij ij   trace Rij R2 trace Q 2 i Algorithm CG-W, we have T 2 TT TT QARBij  ARB j T Ri  trace Rij R2 trace 2 Qi   2 T Ri trace Rij R2 trace Q ij P .  Qi

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22 RR R 2 trace RTT R trace R Rss trace Q P  trace Q P **s sj1  sj22 sj s j trace X Xss1 Q trace X  X s2 Q ss Q QQ Q ss s 2 T 22 R trace P Q RRss2 s jj1 * T trace Q Q  Q  trace X Xss Q22 trace Q ss Q  R s trace Q ss Q 0, 22sj j1  Q Q QQss s j1

R 2 trace PT Q s TTjj1 we obtain 22trace Qsj Q  trace Q sQj1  0, Q Q s j1 T trace Pss1 Q  **  trace X Xss11 Q trace  X X s  1 P s  1 2 Q s Q s and T trace Pss1 Q T trace X** X P trace  X  X Q T ss11 2  ss  1 trace Pss1 Q Q T  s trace Qsj11 Q trace P s2 Q s Q j   TTTTT  Qs AR B AR B ** ss11 trace X Xss11 P trace  X X s  1 T  2  T trace P Q   ss1 T   trace Ps1 Qjsj  2 trace Q Q T Q TT** s BRss11 A X X X X ss  11 BR A  trace  trace P Q trace Q P . 2 sj11  js  which gives TT2 trace C AXss11 B R  trace R sss  111 R  R .

2 R trace QTTT Q trace Q Pj  trace R R  trace R R  By the induction, we know that (5) holds for k  1, 2, . sj11 js2   js  111  js  Qj Theorem 1. Suppose that the matrix equation (1) is consistent 2 nn R and for any initial matrix X1 SAR , the sequence j trace RTT R trace R R 0, 2 sj111  sj  X generated by Algorithm CG-W converges to a solution of Qj  k  (1) after finite-steps.

Thus, (4) holds when ks1. By the induction, we know that Proof: The proof is by contradiction. Assume that Ri  0 ,

(4) holds for when ij , 1,2, , k , ij . imp 1, 2, , , then by Lemma 3, we have Qi  0 * Lemma 3. Suppose that the equation is consistent and X is imp 1, 2, , and can further obtain X mp1 and Rmp1 . If one solution of (1), then the sequences R  and Q  i i Rmp1  0 , then according to Lemma 2 we get the orthogonal generalized by Algorithm CG-W satisfy basis matrix set RR,,, R , R of R , which  12mp mp 1 mp * 2 trace X X Q R, k  1,2, (5) contradicts the assumption. Thus, Rmp1  0 , and X mp1 is the  kk k exact solution of (1).

Proof: We shall also prove this lemma by induction. First of all, Theorem 2. Suppose that the matrix equation (1) is consistent, then we take the initial matrix whenk  1 we have

TT T ** * XP , PAHBBHA2 , trace X X Q trace X  X P trace X  X P 11  1   11  1 1 11

TT TTT ARB ARB ms nn * 11 with any H  R (or specially, for X 0 R ), the trace X X1 1 2  Algorithm CG-W converges to the minimum norm solution of T TT** (1) after finite-steps. BR11 A X X X X 11 BR A  trace  TT T 2 Proof: If we take XP11  , PAHBBHA1   2 with  International Science Index, Mathematical and Computational Sciences Vol:10, No:11, 2016 waset.org/Publication/10006599 ms T ˆ **TT any H  R , by Algorithm CG-W, we can get a solution X AX X11 BR AX X 11 BR  trace of the matrix AXB C after finite-steps, and there exists the 2 matrix Hˆ  Rms , such that XPˆˆ  with * T TT2 trace A X X11 BR trace C AX11 B R trace R 111 R  R ,  PAHBBHAˆˆTTˆ T 2 . From A Xˆˆ X B AXB AXB     Suppose that (5) holds when ks . Owing to we know that all the symmetric arrowhead solution of matrix equation AXB C can be expressed as XXˆ   with X SARnn , satisfying AXB  0 . For XX  , we

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get AXB   0 , thus we have 4 2 AHTTTˆˆ B BHA XXˆˆ,, P X  , X    0 2 HAˆ , X B  0. -2 -4 k thus we get  -6 222 2 XXˆˆ X  X  Xˆ -8

-10 Xˆ is the symmetric arrowhead minimum norm solution of (1). It is not difficult to verify the solution set of (1) is a closed -12 convex set, therefore, the symmetric arrowhead minimum norm -14 solution of (1) is unique. 0 20 40 60 80 100 120 140 160 K III. NUMERICAL EXPERIMENTS Fig. 1 Relation between error  k and iterative number K when i  1 In this section, under the compatibility condition of the constrained matrix equation AXB C , we give an example to In Table I, we obtain iterative steps(Iter), Iterative time (CPU) illustrate the efficiency and investigate the performance of and residual norm RAXBC of the algorithm Algorithm CG-W which has been shown to be numerically kkF reliable in various circumstances. All functions are defined by respectively. We set a stop criterion for RAXBC107 . Then, Fig. 1 plots the relation Matlab 7.0 and all codes are calculated with machine precision kkF around109 . between error  k log10 (AXB C ) and the iterative number Example 1. Given A  toeplitz(1: 30* i ), zeros (30* i ,11* i ) of K when i 1.

row full rank, B  eye(40* i ); ones ( i ,40* i ) of column full We choose the initial matrix Xzerosi0  (41* ,41* i ) , the rank for i  1, 2, 5 and C AXB .Given Y 0.5 ones ( n , n ) unique minimal norm solution of the matrix equation (1) is obtained by the algorithm CG-W. It can be seen from the Table and  XEYPYdiagYEYE 11  1  2. 11 11 Notice that in I, when the order of the matrices A and B is growing this case, the matrix equation C AXB is consistent and has a exponentially, the iterative steps of the algorithm CG-W is unique minimum norm solution. growth multiples.

TABLE I REFERENCES THE ITERATIVE STEPS, ITERATIVE TIME AND RESIDUAL NORM OF THE ALGORITHM CG-W [1] Y.F. Xu, An inverse eigenvalue problem for a special kind of matrices. Math. Appl., 1(1996)68-75. CG-W [2] C.J. Meng, X.Y. Hu, L. Zhang, The skew symmetric orthogonal solution Iter 94 i 1 of the matrix equation AXB=C, Linear Algebra Appl. 402(2005)303-318. CPU 0.282 [3] Li Jiaofen, Zhang Xiaoning, Peng Zhenyun, Alternative projection 3.8626e-008 algorithm for single variable linear constraints matrix equation problems, Rk Mathematica Numerical Since, 36(2014)143-162. Iter 249 [4] Y.X. Peng, X.Y. Hu, L. Zhang, An iteration method for the symmetric i  2 solutions and the optimal approximation solution of the matrix equation CPU 1.690 AXB=C, Applied Mathematics and Computation, 160(2005)763-777. 4.1348e-008 [5] Z.Y. Peng, A matrix LSQR iterative method to solve matrix equation R k AXB=C, International Journal of Computer Mathematics, i  3 Iter 420 87(2010)1820-1830. CPU 7.995 [6] J.F. Li, X.F. Duan, L. Zhang, Numerical solutions of AXB=C for mirror 8.8017e-008 symmetric matrix X under a specified submatrix constraint. Rk Computing, 90(2010) 39-56. International Science Index, Mathematical and Computational Sciences Vol:10, No:11, 2016 waset.org/Publication/10006599 Iter 609 [7] Von Neumann J., Functional Operators. II. The Geometry of Spaces, i  4 Annals of Mathematics Studies, vol.22, Princeton University Press, CPU 23.669 Princeton, 1950. 9.5693e-008 R [8] Cheney W., Goldstein A. Proximity maps for convex sets, Proceedings of k the American Mathematical Society, 10(1959)448-450. Iter 820 i  5 [9] Hongyi Li, Zongsheng Gao, Di Zhao, Least squares solutions of the CPU 62.574 matrix equation with the least norm for symmetric arrowhead matrices. Appl. Math. Comput., 226(2014)719-724. 9.7150e-008 Rk

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