The Relaxed Gradient Based Iterative Algorithm for the Symmetric ( Skew Symmetric ) Solution of the Sylvester Equation AX + XB = C

In this paper, we present two different relaxed gradient based iterative (RGI) algorithms for solving the symmetric and skew symmetric solution of the Sylvester matrix equation AX + XB = C. By using these two iterative methods, it is proved that the iterative solution converges to its true symmetric (skew symmetric) solution under some appropriate assumptions when any initial symmetric (skew symmetric) matrixX0 is taken. Finally, two numerical examples are given to illustrate that the introduced iterative algorithms are more efficient.

As is well known, (1) has a unique solution if and only if  and − possess no common eigenvalues [16] and the solution can be computed by solving a linear system ( ⊗  +   ⊗ )vec () = vec ().Using this method, it will increase the computational cost and storage requirements, so that this approach is only applicable for small sized Sylvester equations.

Mathematical Problems in Engineering
Due to these drawbacks, many other methods for the solution have appeared in the literature.The idea of transforming the coefficient matrix into a Schur or Hessenberg form to compute (1) have been presented in [16,17].When the linear matrix (1) is inconsistent, a finite iterative method to solving its Hermitian minimum norm solutions has been presented in [18].An efficient iterative method based on Hermitian and skew Hermitian splitting has been proposed in [19].Krylov subspace based methods have been presented in [20][21][22][23][24][25][26] for solving Sylvester equations and generalized Sylvester equations.Recently based on the idea of a hierarchical identification principle [27][28][29], some efficient gradient based iterative algorithms for solving generalized Sylvester equations and coupled (general coupled) Sylvester equations have been proposed in [27,[30][31][32].Particularly, for Sylvester equations of form (1), it is illustrated in [33] that the unknown matrix to be identified can be computed by a gradient based iterative algorithm.The convergence properties of the methods are investigated in [27,32].Niu et al. [34] proposed a relaxed gradient based iterative algorithm for solving Sylvester equations.Wang et al. [35] proposed a modified gradient based iterative algorithm for solving Sylvester equations (1).More recently, Xie and Ma [36] gave an accelerated gradient based iterative algorithm for solving (1).In [37,38] Xie et al. studied the special structure solution of matrix equation (1) by using iterative method.
In this paper, inspired by [28,[34][35][36], we first derive a relaxed gradient based iterative (RGI) algorithm for solving the symmetric solution of matrix equation (1).Theoretical analysis shows that our method converges to the exact symmetric solution for any initial value with some appropriate assumptions.Then the proposed algorithm can be also applied to the skew symmetric solution of matrix equation (1).Numerical results illustrate that the proposed method is correct and feasible.We must point out that the ideas in this paper have some differences comparing with that in [28,[34][35][36].
The rest of the paper is organized as follows.In Section 2, some main preliminaries are provided.In Section 3, the relaxed gradient based iterative methods are studied.Finally, a numerical example is included to verify the superior convergence for the algorithms.

Preliminaries
In this section, we reviewed the ideas and principles of the gradient based iterative (GI) method, the relaxed gradient based iterative (RGI) method, and the modified gradient based iterative (MGI) method.
Step 3. Update the sequences Step 5. Update the sequences Step 6. Compute Step 7. Set  fl  + 1; return to Step 2.
More recently, Xie and Ma [36] presented the following AGBI algorithm for solving (1) based on the idea of MGI.

Main Results
In this section, we first study the necessary and sufficient conditions of the symmetric solution for (1).Then the relaxed gradient based iterative algorithm for the symmetric solution of equation ( 1) is proposed.Following the same line, the relaxed gradient based iterative algorithm for the skew symmetric solution of equation ( 1) is also presented.Proof.If   is a unique symmetric solution of (1), then (  )  =   and   +    = ; further we have This shows that   is also the solution of the pair matrix equations (15).Conversely, if the system of matrix equations ( 15) has a common solution  * , let us denote   = ( * + ( * )  )/2; then we can check that This implies that   is the unique symmetric solution of (1).
According to Theorem 6, we construct a relaxed gradient based iterative algorithm to solve the symmetric solution of (1).
Further the sequence {  ()} converges to   , where   is the unique symmetric solution of (1).
Proof.Define the error matrix We have The following two equalities are easily derived.

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Therefore, the above equality implies that It follows from The above two inequalities (25) Following the same line, the idea of Algorithm 7 can be extended to solve the skew symmetric solution of (1).First, we need the following theorem.The relaxed gradient based iterative algorithm for solving the skew symmetric solution of (1) can be stated as follows.
Similarly, we have the following theorem to ensure the convergence of the Algorithm 10.
Theorem 11.Assume the matrix equations (30) have a unique solution  * ; then the iterative sequence {()} generated by the Algorithm 10 converges to  * , if that is, lim →∞ () =  * or the error () −  * converges to zero for any initial value (0).
Furthermore, the sequence {  ()} converges to   , where   is the unique skew symmetric solution of (1).

Numerical Examples
In this section, two numerical examples are used to show the efficiency of the RGI Method.All the computations were performed on Intel5 Core6 i7-4500U CPU @ 1.80 GHZ 2.40 GHZ system by using MATLAB 7.0.EER is the Frobenius norm of absolute error matrices which is defined to be EER = ‖() −  * ‖  , where () is the th iteration result for the RGI Method.