Dykstra ’ s Algorithm for the Optimal Approximate Symmetric Positive Semidefinite Solution of a Class of Matrix Equations *

Dykstra’s alternating projection algorithm was proposed to treat the problem of finding the projection of a given point onto the intersection of some closed convex sets. In this paper, we first apply Dykstra’s alternating projection algorithm to compute the optimal approximate symmetric positive semidefinite solution of the matrix equations AXB = E, CXD = F. If we choose the initial iterative matrix X0 = 0, the least Frobenius norm symmetric positive semidefinite solution of these matrix equations is obtained. A numerical example shows that the new algorithm is feasible and effective.


Introduction
Throughout this paper, we use m n R × and  =  A A A then m n R × is a real Hilbert space.In order to develop this paper, we need to give the following definition.
Definition 1.1.[1] Let M be a closed convex subset in a real Hilbert space H and u be a point in H, then the point in M nearest to u is called the projection of u onto M and denoted by i.e.
n n Obviously, Ω is the symmetric positive semidefinite solution set of the matrix equations (1.3).It is easy to verify that Ω is a closed convex set, then the solution X of Problem I is unique.In this paper, the unique solution X is called the optimal approximate symmetric positive semidefinite solution of Equation (1.3).In particular, if 0, = X then the solution X of Problem I is just the least Frobenius norm symmetric positive semidefinite solution of the matrix equations (1.3).
This kind of matrix nearness problem occurs frequently in experimental design, see for instance [2] [3].Here X may be obtained from experiments, but not satisfy Equation (1.3).The nearest matrix X satisfies Equation (1.3) and is nearest to the given matrix X .Up to now, Equation (1.3) and their matrix nearness problem I have been extensively studied for the past 40 or more years.Navarra-Odell-Young [4] and Wang [5] gave necessary and sufficient conditions for Equation (1.3) having a solution and presented the expression for a general solution.By the projection theorem and matrix decompositions, Liao-Lei- Yuan [6] [7] gave some analytical expressions of the optimal approximate least square symmetric solution of Equation (1.3).Sheng-Chen [8] presented an efficient iterative method to compute the optimal approximate solution for the matrix equations (1.3).Ding-Liu-Ding [9] considered the unique solution of Equation (1.3) and used gradient based iterative algorithm to compute the unique solution.Peng-Hu-Zhang [10] and Chen-Peng-Zhou [11] proposed some iterative methods to compute the symmetric solutions and optimal approximate symmetric solution of Equation (1.3).The (least square) solution and the optimal approximate (least square) solution of Equation (1.3), which is constrained as bisymmetric, reflexive, generalized reflexive, generalized centro-symmetric, were studied in [11]- [17].Nevertheless, to the best of our knowledge, the optimal approximate solution of Equation (1.3), which is constrained as symmetric positive semidefinite, (i.e.Problem I) has not been solved.The difficulty of Problem I lies in how to characterize the convex set Ω .In this paper, we first divided the set Ω into three sets 1 , Ω 2 , Ω 3 , Ω and then adopt alternating projections to overcome the difficulty.Dykstra's alternating projection algorithm was proposed by Dykstra [18] to treat the problem of finding the projection of a given point onto the intersection of some closed convex sets.It is based on a clear modification of the classical alternating projection algorithm first proposed by Von Neumann [19], and studied later by Cheney and Goldstein [20].For an application of Dykstra's alternating projection algorithm to compute the nearest diagonally dominant matrix see [21].For a complete survey on Dykstra's alternation projection algorithm and applications see Deutsch [22].
In this paper, we propose a new algorithm to compute the optimal approximate symmetric positive semidefinite solution of Equation (1.3).We state Problem I as the minimization of a convex quadratic function over the intersection of three closed convex sets in the vector space .
n n R × From this point of view, Problem I can be solved by the Dykstra's alternating projection algorithm.If we choose the initial iterative matrix 0 0, = X the least Frobenius norm symmetric positive semidefinite solution of the matrix equations , = = AXB E CXD F is obtained.In the end, we use a numerical example to show that the new algorithm is feasible and effective.

Dykstra's Algorithm for Solving Problem I
In this section, we apply Dykstra's alternating projection algorithm to compute the optimal approximate symmetric positive semidefinite solution of Equation (1.3).We first introduce Dykstra's alternating projection algorithm and its convergence theorem.
In order to find the projection of a given point onto the intersection of a finite number of closed convex sets 1 2 , , , , Dykstra [18] proposed Dykstra's alternating projection algorithm which can be stated as follows.This algorithm can be also seen in [1] [23]- [25].
Dykstra's Algorithm 2.1 1) Given the initial value 0 The utility of Dykstra's algorithm 2.1 is based on the following theorem (see [23]- [25] and the references therein).
Lemma x generated by Dykstra's algorithm 2.1 converge to ( ) .
Now we begin to use Dykstra's algorithm 2.1 to solve Problem I. Firstly, we define three sets , and . .
On the other hand, it is easy to verify that 1 , Ω 2 Ω and 3 Ω are closed convex subsets of the real Hilbert space n n R × .After defining the sets 1 , Ω 2 Ω and 3 Ω , Problem I can be rewritten as finding By Definition 1.1 and noting that the equalities (2.2) and (1.2), it is easy to find that ( ) 4 Therefore, Problem I can be converted equivalently into finding the projection ( ) . Now we will use Dykstra's algorithm 2.1 to compute the projection ( ) . By (2.3), we can get the optimal approximate symmetric positive semidefinite solution X of the matrix equations (1.3).
We can see that the key problems to realize Dykstra's algorithm 2.1 are how to compute the projections ( ) ( ) 3 Ω , respectively.Such problems are perfectly solvable in the following theorems.
Theorem 2.1.Suppose that the set 1 Ω is nonempty.For a given n n × matrix Z, we have ( ) ( ) Proof.By Definition 1.1, we know that the projection ( ) Now we begin to solve the minimization problem (2.4).We first characterize the solution set 1 , Ω and then find X such that (2.4) holds.Noting that the set 1 Ω is a closed convex set, then the minimization problem (2.4) has a unique solution.Hence ( ) The singular value decomposition of the matrices A and B are given by , , , , , , , , , , , , s diag w w w = W  .According to the definition of the Moore-Penrose generalized inverse of a matrix, we have , , , .
Then the matrix equation = AXB E can be equivalently written as .
Ω is nonempty, by (2.5) it is easy to verify that (2.9), (2.10) and (2.11) are identical equations.Hence the general solutions of the matrix equation = AXB E can be expressed as , , , are arbitrary, which implies that the entries of the set 1 Ω can be stated as (2.12).Consequently, , , .
( ) End End By Lemma 2.1 and (2.1), and noting that 1 , Ω 2 Ω and 3 Ω are closed convex sets, we get the convergence theorem for Algorithm 2.2.

Numerical Experiments
In this section, we give a numerical example to illustrate that the new algorithm is feasible and effective to compute the optimal approximate symmetric positive semidefinite solution of the matrix equation (1.3) Here we use ( ) and its residual error 10 41 41 Error 9.67 10 .

− ≈ − + − = × E AX B F CX D
By concrete computations, we know that the distance from X to the solution set Ω is 41 ˆ5.3852.min and its residual error 10 88 88 Error 9.95 10 .

≈ = X X
, 0000 1.0000 1.0000 1.0000 which is also the least Frobenius norm symmetric positive semidefinite solution of the matrix equations (1.3), and its residual error

Conclusion
In this paper, we state Problem I as the minimization of a convex quadratic function over the intersection of three closed convex sets in the Hilbert space n n R × , then we can use Dykstra's alternating projection algorithm to compute the optimal approximate symmetric positive semidefinite solution of the matrix equations (1.3).If we choose the initial matrix 0 0, = X the least Frobenius norm symmetric positive semidefinite solution of the matrix equations (1.3) can be obtained.A numerical example show that the new algorithm is feasible and effective to compute the optimal approximate symmetric positive semidefinite solution of the matrix equations (1.3).
stand for the set of m n × real matrices and n n × symmetric positive semidefinite matrices, respectively.We denote the transpose and Moore-Penrose generalized inverse of the matrix A by T A and + A , respectively.The symbol n I stands for n n × identity matrix.For C. M. Li et al. the inner product of the matrix A and B. The induced norm is the so-called Frobenius norm, that is,

Theorem 2 . 5 .
If the set Ω is nonempty, then the matrix sequences { } 2.2 converge to optimal approximate symmetric positive semidefinite solution X of the matrix equations(1.3).Moreover, if the initial matrix

ˆ6
By concrete computations, we know that the distance from X to the solution set Ω is 116 that Algorithm 2.2 is feasible and effective to compute the optimal approximate symmetric positive semidefinite solution of the matrix equations (1.3).
in Theorems 2.1, 2.2 and 2.3, we get a new algorithm to compute the optimal approximate symmetric positive semidefinite solution X of the matrix equations (1.3) which can be stated as follows.