Journal of Computational and Applied Mathematics. New matrix iterative methods for constraint solutions of the matrix

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1 Journal of Computational and Applied Mathematics 35 ( Contents lists aailable at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: New matrix iteratie methods for constraint solutions of the matrix equation AX = C Zhen-yun Peng Guilin Uniersity of Electronic Technology, Guilin , PR China a r t i c l e i n f o a b s t r a c t Article history: Receied 5 March 007 Receied in reised form 9 August 009 Keywords: Iteratie algorithm Matrix equation Matrix nearness problem Minimum residual problem In this paper, two new matrix iteratie methods are presented to sole the matrix equation AX = C, the minimum residual problem min X S AX C and the matrix nearness problem min X SE X X, where S is the set of constraint matrices, such as symmetric, symmetric R-symmetric and (R, S-symmetric, and S E is the solution set of aboe matrix equation or minimum residual problem. These matrix iteratie methods hae faster conergence rate and higher accuracy than the matrix iteratie methods proposed in Deng et al. (006 [13], Huang et al. (008 [15], Peng (005 [16] and Lei and Liao (007 [17]. Paige s algorithms are used as the frame method for deriing these matrix iteratie methods. Numerical examples are used to illustrate the efficiency of these new methods. 010 Elseier.V. All rights resered. 1. Introduction Denoted by R m n and SR n n the set of m n real matrices and n n symmetric matrices, respectiely. Denoted by R(A, N(A and A the column space, null space and Frobenius norm of the matrix A, respectiely. We use A to stand for the Kronecker production of the matrix A and. For X = [x 1, x,..., x n ] R m n, we denote by ec(x = [x T 1,..., xt n ]T the ector expanded by columns of X. We reconsider numerical methods to compute constraint solutions X of the following problems: AX = C, X S min AX C X S min X X X S E where A R m n, R n p, C R m p and X R n n, S is the set of constraint matrices, such as symmetric, skew symmetric, symmetric R-symmetric, symmetric R-skew symmetric [1,], (R, S-symmetric or (R, S-skew symmetric [3], and S E is the solution set of the matrix equation (1.1 or the minimum residual problem (1.. Direct methods to sole aboe problems with unknown matrix X in arious special structures hae been used by seeral authors. For example, the symmetric solutions were considered in [4 8]. The bisymmetric, centro-symmetric and Renonnegatie definite solutions were considered in [9 1], respectiely. y using direct methods, the necessary and sufficient conditions for the existence of the general solution were deried, and the solution s expressions were gien too. Direct methods, howeer, may be less efficient for large sparse coefficient matrices A and due to limited by storage space and computing speed of the computers. Therefore, iteratie methods to sole matrix equation hae attracted much interests (1.1 (1. (1.3 Research supported by National Natural Science Foundation of China ( , and Proincial Natural Science Foundation of Guangxi ( address: yunzhenp@163.com /$ see front matter 010 Elseier.V. All rights resered. doi: /j.cam

2 Z.-y. Peng / Journal of Computational and Applied Mathematics 35 ( recently. For example, Deng et al. [13] and Peng et al. [14] designed a similar matrix-form iteratie method, according to the fundamental idea of the conjugate gradient method for the standard system of linear equations, to compute the symmetric solutions of the problems (1.1 and (1.3. With the same idea, Huang et al. [15] designed a matrix-form iteratie method to compute the skew symmetric solutions of the problems (1.1 and (1.3. Peng [16] and Liao et al. [17] designed respectie matrix-form iteratie method to compute the symmetric solutions of the problems (1. and (1.3. The matrix-form iteratie methods proposed in [13 15] cannot used to compute the minimum Frobenius norm solution of the problem (1.. The matrix-form iteratie methods proposed in [16,17] can be used to compute the minimum Frobenius norm solutions of both problem (1.1 and problem (1.. ut, in general, the matrix-form iteratie method proposed in [16] has ery slow conergence rate and the matrix-form iteratie method proposed in [17] has ery low accuracy. In this paper, two new matrix iteratie methods are presented to sole the matrix equation (1.1, the minimum residual problem (1. and the matrix nearness problem (1.3. These new matrix iteratie methods hae faster conergence rate and higher accuracy than the iteratie methods proposed in aboe references in some cases. We will use Paige s algorithms [18], which are based on the bidiagonalization procedure of Golub and Kahan [19], as the frame method for deriing these new matrix iteratie methods. The basic idea is that we first characterize constraints matrix X S, then, by the Kronecker production of matrices, we transform problems (1.1 and (1. into the unconstrained linear systems and linear least squares problem in ector form and hence can be soled by Paige s algorithms, and finally, we transform ector form iteratie methods into matrix form iteratie methods. This paper is organized as follows. In Section, we shortly reiew Paige s algorithms to sole linear systems and least squares problem. ased on Paige s algorithms, we propose two new matrix iteratie algorithms to sole problems (1.1 (1.3 in Section 3. Finally, seeral numerical examples are gien to illustrate the efficiency of these new algorithms.. Paige s algorithms Paige s algorithms were used to compute the minimum norm solution x of the following problems: Linear systems: Mx = f Linear least squares: min Mx f (. where. denote the l -norm, that is u = u, u = u T u. Paige s algorithms are based on the bidiagonalization procedure of Golub and Kahan [19] which hae two forms as follows. idiag 1 (starting ector f ; reduction to lower bidiagonal form: β 1 u 1 = f, α 1 1 = M T u 1 }. β i+1 u i+1 = M i α i u i α i+1 i+1 = M T, i = 1,,.... u i+1 β i+1 i The scalars α i 0 and β i 0 are chosen so that u i = i = 1. With the definitions U k = [u 1, u,..., u k ], V k = [ 1,,..., k ], α 1 β α. k = β.. 3,... αk β k+1 the recurrence relations (.3 may be rewritten as U k+1 (β 1 e 1 = f, MV k = U k+1 k, M T U k+1 = V k T k + α k+1 k+1 e T k+1, where e 1 and e k+1 are respectiely the first and last column of the identity matrix I with size implied by context. If exact arithmetic were used, then U T k+1 U k+1 = I and V T k V k = I. idiag (starting ector M T f ; reduction to upper bidiagonal form: θ 1 1 = M T f, ρ 1 p 1 = M } 1. θ i+1 i+1 = M T p i ρ i i, i = 1,,.... ρ i+1 p i+1 = M i+1 θ i+1 p i The scalars ρ i 0 and θ i 0 are chosen so that p i = i = 1. In this case, if ρ 1 θ ρ θ 3 P k = [p 1, p,..., p k ], R V k = [ 1,,..., k ], k = ρ k 1 θ, k ρ k (.1 (.3 (.4

3 78 Z.-y. Peng / Journal of Computational and Applied Mathematics 35 ( the recurrence relations (.4 may be rewritten as V k (θ 1 e 1 = M T f, MV k = P k R k, M T P k = V k R T k + θ k+1 k+1 e T k, and also P T k P k = V T k V k = I if exact arithmetic were used. Applying on the idiag 1 and, Paige constructs two algorithms, which are respectie named Paige algorithm 1 and Paige algorithm, to compute the unique minimum l -norm solution x of the linear system (.1 and the linear least squares (. as follows. Paige algorithm 1 (1 τ 0 = 1; ξ 0 = 1; ω 0 = 0; z 0 = 0; w 0 = 0; β 1 u 1 = f ; α 1 1 = M T u 1 ; ( For i = 1,,... until {x i } conergence, do (a ξ i = ξ i 1 β i /α i ; z i = z i 1 + ξ i i ; (b ω i = (τ i 1 β i ω i 1 /α i ; w i = w i 1 + ω i i ; (c β i+1 u i+1 = M i α i u i ; (d τ i = τ i 1 α i /β i+1 ; (e α i+1 i+1 = M T u i+1 β i+1 i ; (f γ i = β i+1 ξ i /(β i+1 ω i τ i ; (g x i = z i γ i w i. Paige algorithm (1 θ 1 1 = M T f ; ρ 1 p 1 = M 1 ; w 1 = 1 /ρ 1 ; ξ 1 = θ 1 /ρ 1 ; x 1 = ξ 1 w 1 ; ( For i = 1,,... until {x i } conergence, do (a θ i+1 i+1 = M T p i ρ i i ; (b ρ i+1 p i+1 = M i+1 θ i+1 p i ; (c w i+1 = ( i+1 θ i+1 w i /ρ i+1 ; (d ξ i+1 = ξ i θ i+1 /ρ i+1 ; (e x i+1 = x i + ξ i+1 w i+1. The scalars α i 0, β i 0 in Paige algorithm 1 and the scalars ρ i 0, θ i 0 in Paige algorithm are chosen so that u i = i = p i = i = 1. Paige also pointed out that the stopping criteria on Paige algorithms 1 and can be used as f Mx i ε, ξ i ε or x i x i 1 ε, where ε > 0 is a small tolerance, to compute the unique minimum l -norm solution of the linear equations (.1 and as M T (f Mx i ε, or x i x i 1 ε to compute the unique minimum l -norm solution of the linear least square (.. 3. New matrix iteratie methods ased on Paige algorithms 1 and, we propose two new matrix iteratie algorithms to sole (1.1 (1.3 in this section. We first consider the linear matrix equation (1.1 and the minimum residual problem (1. with unknown matrix X is following three cases (if matrix X is other case, the analogous results can be obtained, and thus is omitted here. Case 1. X is a symmetric matrix. Case. X is a symmetric R-symmetric matrix, that is, X {Y R n n Y = Y T, RYR = Y, R T = R = R 1 R n n }. Case 3. X is a (R, S-symmetric matrix, that is, X {Y R m n RYS = Y, R T = R = R 1 R m m, S T = S = S 1 R n n }. In case 1. Noting that X is the symmetric solution of the matrix equation AX = C if and only if X is the symmetric solution of the system of matrix equations { AX = C T XA T = C T. (3.1 And the system of matrix equations (3.1 can be transformed into the system of linear equations (.1 with coefficient matrix M and ector f as M =, A T f = ec(c T.

4 Z.-y. Peng / Journal of Computational and Applied Mathematics 35 ( Therefore, β 1 u 1 = f, α 1 1 = M T u 1, β i+1 u i+1 = M i α i u i (i = 1,,..., and α i+1 i+1 = M T u i+1 β i+1 i (i = 1,,..., can be written as β 1 u 1 = ec(c T, (3. α 1 1 = ( A T, A T u 1, β i+1 u i+1 = A T i α i u i, α i+1 i+1 = ( A T, A T u i+1 β i+1 i. (3.5 From (3. (3.5, we hae ec(ui u i = ec(u T, i i = ec(v i, U i R m p, V i SR n n. And so, the ector form β 1 u 1 = f, α 1 1 = M T u 1, β i+1 u i+1 = M i α i u i (i = 1,,..., and α i+1 i+1 = M T u i+1 β i+1 i (i = 1,,..., in Paige algorithm 1 can be rewritten as matrix form β 1 U 1 = C, β 1 = C, α 1 V 1 = A T U 1 T + U T A, α 1 1 = A T U 1 T + U T A, 1 β i+1 U i+1 = AV i α i U i, β i+1 = AV i α i U i, α i+1 V i+1 = A T U i+1 T + U T i+1 A β i+1v i, α i+1 = A T U i+1 T + U T i+1 A β i+1v i. Analogously, θ 1 1 = M T f, ρ 1 p 1 = M 1, θ i+1 i+1 = M T p i ρ i i (i = 1,,..., and ρ i+1 p i+1 = M i+1 θ i+1 p i (i = 1,,..., can be written as θ 1 1 = ( A T, A T ec(c T, (3.6 ρ 1 p 1 = A T 1, (3.7 θ i+1 i+1 = ( A T, A T p i ρ i i, ρ i+1 p i+1 = A T i+1 ρ i+1 p i. From (3.6 (3.9, we hae i = ec(v i, p i = ec(pi ec(p T, V i i SR n n, P i R m p. And so, the ector form θ 1 1 = M T f, ρ 1 p 1 = M 1, θ i+1 i+1 = M T p i ρ i i (i = 1,,..., and ρ i+1 p i+1 = M i+1 θ i+1 p i (i = 1,,..., in Paige algorithm can be rewritten as matrix form θ 1 V 1 = (A T C T + C T A, ρ 1 P 1 = AV 1, ρ 1 = AV 1, θ i+1 V i+1 = A T P i T + P T i A ρ i V i, θ i+1 = A T P i T + P T i A ρ i V i, ρ i+1 P i+1 = AV i+1 θ i+1 P i, ρ i+1 = AV i+1 θ i+1 P i. θ 1 = A T C T + C T A, In case. Noting that X is the symmetric R-symmetric solution of the matrix equation AX = C if and only if X is the symmetric R-symmetric solution of the system of matrix equations AX = C T XA T = C T (3.10 ARXR = C T RXRA T = C T. (3.3 (3.4 (3.8 (3.9

5 730 Z.-y. Peng / Journal of Computational and Applied Mathematics 35 ( And the system of matrix equations (3.10 can be transformed into the system of linear equations (.1 with coefficient matrix M and ector f as T A M = A T T R AR, f = ec(c T. AR T R ec(c T Therefore, β 1 u 1 = f, α 1 1 = M T u 1, β i+1 u i+1 = M i α i u i (i = 1,,..., and α i+1 i+1 = M T u i+1 β i+1 i (i = 1,,..., can be written as ec(c β 1 u 1 = T, (3.11 ec(c T α 1 1 = ( A T, A T, R RA T, RA T R u 1, (3.1 β i+1 u i+1 = A T T R AR i α i u i, (3.13 AR T R α i+1 i+1 = ( A T, A T, R RA T, RA T R u i+1 β i+1 i. (3.14 From (3.11 (3.14, we hae ec(u i ec(u u i = T i ec(u i, i = ec(v i, ec(u T i where U i R m p, V i is a symmetric R-symmetric matrix. And so, the ector form β 1 u 1 = f, α 1 1 = M T u 1, β i+1 u i+1 = M i α i u i (i = 1,,..., and α i+1 i+1 = M T u i+1 β i+1 i (i = 1,,..., in Paige algorithm 1 can be rewritten as matrix form β 1 U 1 = C, β 1 = C, α 1 V 1 = A T U 1 T + U T 1 A + R(AT U 1 T + U T AR, 1 α 1 = A T U 1 T + U T 1 A + R(AT U 1 T + U T 1 AR, β i+1 U i+1 = AV i α i U i, β i+1 = AV i α i U i, α i+1 V i+1 = A T U i+1 T + U T i+1 A + R(AT U i+1 T + U T i+1 AR β i+1v i, α i+1 = A T U i+1 T + U T i+1 A + R(AT U i+1 T + U T i+1 AR β i+1v i. Analogously, θ 1 1 = M T f, ρ 1 p 1 = M 1, θ i+1 i+1 = M T p i ρ i i (i = 1,,..., and ρ i+1 p i+1 = M i+1 θ i+1 p i (i = 1,,..., can be written as θ 1 1 = ( A T, A T, R RA T, RA T ec(c T R, (3.15 ec(c T ρ 1 p 1 = A T T R AR 1, (3.16 AR T R θ i+1 i+1 = ( A T, A T, R RA T, RA T R p i ρ i i, (3.17 ρ i+1 p i+1 = A T T R AR i+1 ρ i+1 p i. (3.18 AR T R

6 Z.-y. Peng / Journal of Computational and Applied Mathematics 35 ( From (3.15 (3.18, we hae ec(p i ec(p i = ec(v i, p i = T i ec(p i, ec(p T i where P i R m p, V i is a symmetric R-symmetric matrix. And so, the ector form θ 1 1 = M T f, ρ 1 p 1 = M 1, θ i+1 i+1 = M T p i ρ i i (i = 1,,..., and ρ i+1 p i+1 = M i+1 θ i+1 p i (i = 1,,..., in Paige algorithm can be rewritten as matrix form θ 1 V 1 = A T C T + C T A + R(A T C T + C T AR, θ 1 = A T C T + C T A + R(A T C T + C T AR, ρ 1 P 1 = AV 1, ρ 1 = AV 1, θ i+1 V i+1 = A T P i T + P T i A + R(A T P i T + P T i AR ρ i V i, θ i+1 = A T P i T + P T i A + R(A T P i T + P T i AR ρ i V i, ρ i+1 P i+1 = AV i+1 θ i+1 P i, ρ i+1 = AV i α i U i. In case 3. Noting that X is the (R, S-symmetric solution of the matrix equation AX = C if and only if X is the (R, S- symmetric solution of the system of matrix equations { AX = C ARXS = C. (3.19 And the system of matrix equations (3.19 can be transformed into the system of linear equations (.1 with coefficient matrix M and ector f as M = ( T S AR, f =. Therefore, β 1 u 1 = f, α 1 1 = M T u 1, β i+1 u i+1 = M i α i u i (i = 1,,..., and α i+1 i+1 = M T u i+1 β i+1 i (i = 1,,..., can be written as β 1 u 1 =, (3.0 α 1 1 = ( A T, S RA T u 1, β i+1 u i+1 = T S i α i u i, AR (3.1 (3. α i+1 i+1 = ( A T, S RA T u i+1 β i+1 i. (3.3 From (3.0 (3.3, we hae ec(ui u i =, ec(u i i = ec(v i, where U i R m p, V i is a (R, S-symmetric matrix. And so, the ector form β 1 u 1 = f, α 1 1 = M T u 1, β i+1 u i+1 = M i α i u i (i = 1,,..., and α i+1 i+1 = M T u i+1 β i+1 i (i = 1,,..., in Paige algorithm 1 can be rewritten as matrix form β 1 U 1 = C, β 1 = C, α 1 V 1 = A T U 1 T + RA T U 1 T S, β i+1 U i+1 = AV i α i U i, β i+1 = AV i α i U i α i+1 V i+1 = A T U i+1 T + RA T U i+1 T S β i+1 V i, α i+1 = A T U i+1 T + RA T U i+1 T S β i+1 V i. α 1 = A T U 1 T + RA T U 1 T S,

7 73 Z.-y. Peng / Journal of Computational and Applied Mathematics 35 ( Analogously, the formula θ 1 1 = M T f, ρ 1 p 1 = M 1, θ i+1 i+1 = M T p i ρ i i (i = 1,,..., and ρ i+1 p i+1 = M i+1 θ i+1 p i (i = 1,,..., can be written as θ 1 1 = ( A T, S RA T, (3.4 ρ 1 p 1 = T S 1, (3.5 AR θ i+1 i+1 = T p S i ρ i i, (3.6 AR ρ i+1 p i+1 = A T i+1 ρ i+1 p i. (3.7 From (3.4 (3.7, we can obtain that ec(pi i = ec(v i, p i =, ec(p i where P i R m p, V i is a (R, S-symmetric matrix. And so, the ector form θ 1 1 = M T f, ρ 1 p 1 = M 1, θ i+1 i+1 = M T p i ρ i i (i = 1,,..., and ρ i+1 p i+1 = M i+1 θ i+1 p i (i = 1,,..., in Paige algorithm can be rewritten as matrix form θ 1 V 1 = A T C T + RA T C T S, ρ 1 P 1 = AV 1, ρ 1 = AV 1, θ i+1 V i+1 = A T P i T + RA T P i T S ρ i V i, θ i+1 = A T P i T + RA T P i T S ρ i V i, ρ i+1 P i+1 = AV i+1 θ i+1 P i, ρ i+1 = AV i+1 θ i+1 P i. θ 1 = A T C T + RA T C T S, Analogous results can be obtained about the minimum residual problem (1.. According to aboe discussion, we can design two matrix form iteratie methods to compute the unique minimum Frobenius norm solution X of the linear matrix equation (1.1 and the minimum residual problem (1.. As an example when unknown matrix X is constrained as symmetric, the matrix form iteratie methods may be listed as following Paige1_M and Paige_M. Paige1_M (1 τ 0 = 1; ξ 0 = 1; ω 0 = 0; Z 0 = 0; W 0 = 0; β 1 U 1 = C; β 1 = C ; α 1 V 1 = A T U 1 T + U T 1 A; α 1 = A T U 1 T + U T 1 A ; ( For i = 1,,... until {X i } conergence, do (a ξ i = ξ i 1 β i /α i ; Z i = Z i 1 + ξ i V i ; (b ω i = (τ i 1 β i ω i 1 /α i ; W i = W i 1 + ω i V i ; (c β i+1 U i+1 = AV i α i U i ; β i+1 = AV i α i U i ; (d τ i = τ i 1 α i /β i+1 ; (e α i+1 V i+1 = A T U i+1 T + U T i+1 A β i+1v i ; α i+1 = A T U i+1 T + U T i+1 A β i+1v i ; (f γ i = β i+1 ξ i /(β i+1 ω i τ i ; (g X i = Z i γ i W i. Paige _M (1 θ 1 V 1 = (A T C T + C T A; θ 1 = A T C T + C T A ; ρ 1 P 1 = AV 1 ; ρ 1 = AV 1 ; W 1 = V 1 /ρ 1 ; ξ 1 = θ 1 /ρ 1 ; X 1 = ξ 1 W 1 ; ( For i = 1,,... until {X i } conergence, do (a θ i+1 V i+1 = A T P i T + P T i A ρ i V i ; θ i+1 = A T P i T + P T i A ρ i V i ; (b ρ i+1 P i+1 = AV i+1 θ i+1 P i ; ρ i+1 = AV i+1 θ i+1 P i ;

8 Z.-y. Peng / Journal of Computational and Applied Mathematics 35 ( (c W i+1 = (V i+1 θ i+1 W i /ρ i+1 ; (d ξ i+1 = ξ i θ i+1 /ρ i+1 ; (e X i+1 = X i + ξ i+1 W i+1. The stopping criteria on the algorithms Paige1_M and Paige_M can be used as C AX i ε, ξ i ε or X i X i 1 ε, where ε > 0 is a small tolerance, to compute the unique minimum Frobenius norm solution of the linear matrix equation (1.1 and as A T C T + C T A A T AX i T T X i A T A ε or X i X i 1 ε to compute the unique minimum Frobenius norm solution of the minimum residual problem (1.. Noting that X i obtained in Paige1_M and Paige_M is a symmetric matrix, we know that X i is the symmetric solution of matrix equation (1.1 when X i is the solution of the system of matrix equations (3.1. Now we consider the matrix nearness problem (1.3. We only discuss X S E is a symmetric matrix. Noting that, for arbitrary matrix X R n n, it follows min X X = min X X T + X + X T X. X SR n n X SR n n Hence, finding the unique symmetric solution of the matrix nearness problem (1.3 is equialent to first find the minimum Frobenius norm symmetric solution of the matrix equation (1.1 or the minimum residual problem (1. with C A X+ XT instead of C. Once the minimum Frobenius norm symmetric solution X is obtained by Paige1_M and Paige_M, the unique symmetric solution ˆX of the matrix nearness problem (1.3 can be obtained. In this case, the solution ˆX can be expressed as ˆX = X + X+ XT. Analogously, if the matrix nearness problem (1.3 with unknown matrix X S E is a skew symmetric, symmetric R-symmetric, symmetric R-skew symmetric, (R, S-symmetric or (R, S-skew symmetric, matrix C in the linear matrix equation (1.1 or the minimum residual problem (1. substitute, respectiely, by C A X XT C A X+ XT R( X+ XT R here. 4. Numerical examples 4, C A X+R XS, C A X+ XT +R( X+ XT R, 4 or C A X R XS. The following process are the same as aboe, and thus is omitted In this section, we compare Paige1_M and Paige_M numerically with three methods proposed in [13,16,17], denoted, respectiely, by Deng_M, Peng_M and Liao_M. All the tests were performed by MATLA 7.1 and the initial iteratie matrices in the methods Deng_M, peng_m and Liao_M are chosen as zero matrices in suitable size. All the following examples are used to illustrate the performance of fie methods to compute the minimum [ Frobenius ] norm symmetric solution X of the matrix equation (1.1 and the minimum residual problem (1.. Let M = A T. We use κ(m to stand for the spectral condition number of M, that is, κ(m = σ 1δ 1 σ r δ r, where σ 1 and σ r are respectiely the maximum and minimum nonzero singular alues of A, and δ 1 and δ r are respectiely the maximum and minimum nonzero singular alues of. Example 4.1. Choose arbitrary random matrices A = rand(5, 6 and = rand(6, 7 (in Matlab notation, such as A = = , then κ(m 58.. Let C = Aones(6, 6, then the matrix equation AX = C is consistent, and hence has a unique minimum Frobenius norm solution. Let C = ones(5, 7, then the matrix equation AX = C is inconsistent, and hence has a unique minimum Frobenius norm least squares solution. Fig. 1 illustrates the performance of the methods: Deng_M, Peng_M, Liao_M, Paige1_M and Paige_M in case C = Aones(6. Fig. illustrates the performance of the methods: Peng_M, Liao_M, Paige1_M and Paige_M in case C = ones(5, 7. Example 4.. Suppose that the matrices A, and C be gien by example 5.3 in [17]: A is a real n n(n = l + 6 blockdiagonal matrix with first l blocks are the form i ai b and the last two blocks are and, is b i a i ( (

9 734 Z.-y. Peng / Journal of Computational and Applied Mathematics 35 ( Fig. 1. Conergence cures of the function h(x k = log 10 C AX k. Fig.. Conergence cures of the function h(x k = log 10 A T C T + C T A A T AX k T T X k A T A. Fig. 3. Conergence cures of the function h(x k = log 10 C AX k. ( e d 0 a real n n block-diagonal matrix with the 3 3 block d e d. We consider the following two cases 0 d e Case I: n = 96, a i = i, b i = a i (1 i l, d =, e = 0, κ(m = 305, C = Aones(n, n. Case II: n = 96, a i =, b i = a i (1 i l, d =, e =, κ(m = 144, C = ones(n, n.

10 Z.-y. Peng / Journal of Computational and Applied Mathematics 35 ( Fig. 4. Conergence cures of the function h(x k = log 10 A T C T + C T A A T AX k T T X k A T A. Obiously, the matrix equation AX = C has a symmetric solution in case I and has no symmetric solution in case II. Fig. 3 illustrates the performance of the methods: Deng_M, Peng_M, Liao_M, Paige1_M and Paige_M in case I. Fig. 4 illustrates the performance of the methods: Peng_M, liao_m, Paige1_M and Paige_M in case II. The aboe two examples and many other examples we hae tested by MATLA confirm the following conergence results of the fie iterations: Deng_M is quite efficient to sole linear matrix equation (1.1, but do not suitable for soling minimum residual problem (1.. Liao_M is efficient when A and are sparse matrices with small spectral condition number. When A and are dense matrix with high spectral condition number, Liao_M has ery low accuracy. Peng_M, in general, has higher accuracy and ery slow conergence rate. Paige1_M has faster conergence rate and higher accuracy than other method to sole the linear matrix equation (1.1. Paige_M has faster conergence rate and higher accuracy than other methods to sole the minimum residual problem (1.. References [1] William F. Trench, Hermitian, hermitian R-symmetric, and hermitian R-skew symmetric Procrustes problems, Linear Algebra Appl. 387 ( [] Z.Y. Peng, X.Y. Hu, The reflexie and anti-reflexie solutions of the matrix equation AX =, Linear Algebra Appl. 375 ( [3] William F. Trench, Minimization problem for (R, S-symmetric and (R, S-skew symmetric matrices, Linear Algebra Appl. 389 ( [4] H. Dai, On the symmetric solutions of linear matrix equations, Linear Algebra Appl. 131 ( [5] H. Dai, On the symmetric solutions of linear matrix equations, Linear Algebra Appl. 39 ( [6] K.E. Chu, Symmetric solutions of linear matrix equations by matrix decompositions, Linear Algebra Appl. 119 ( [7] A.P. Liao, Y. Lei, Optimal approximate solution of the matrix equation AX = C oer symmetric matrices, J. Comput. Math. 5 ( [8] A.P. Liao, Z.Z. ai, Least-squares solution of AX = D oer symmetric positie semidefinite matrices X, J. Comput. Math. 1 ( [9] A.P. Liao, Z.Z. ai, Least-squares solutions of the matrix equation A XA = D in bisymmetric matrix set, Math. Numer. Sinica 4 ( [10] A.P. Liao, Z.Z. ai, The constrained solutions of two matrix equations, Acta Math. Sinica (English Ser. 18 ( [11] Z.Y. Peng, The centro-symmetric solutions of linear matrix equation AX = C and its optimal approximation, J. Eng. Math. 6 ( [1] Q.W. Wang, C.L. Yang, The re-nonnegatie definite solutions to the matrix equation AX = C, Comment. Math. Uni. Carolinae 39 ( [13] Y.. Deng, Z.Z. ai, Y.H. Gao, Iteratie orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations, Numer. Linear Algebra Appl. 13 ( [14] X.Y. Peng, X.Y. Hu, L. Zhang, An iteration method for the symmetric solutions and the optomal approximation solution of matrix equation AX = C, Appl. Math. Comput. 160 ( [15] G.X. Huang, F. Yin, K. Guo, An iteratie method for the skew-symmetic solution and the optimal approximate solution of the matrix equation AX = C, J. Comput. Appl. Math. 1 ( [16] Z.Y. Peng, An iteratie method for the least squares symmetric solution of the linear matrix equation AX = C, Appl. Math. Comput. 170 ( [17] Y. Lei, A.P. Liao, A minimal residual algorithm for the inconsistent matrix equation AX = C oer symmetric matrices, Appl. Math. Comput. 188 ( [18] C.C. Paige, idiagonalization of matrices and solution of linear equation, SIAM. J. Numer. Anal. 11 ( [19] G.H. Golub, W. Kahan, Calculating the singular alues and pseudoinerse of a matrix, SIAM J. Numer. Anal. (