(R, S) conjugate solution to coupled Sylvester complex matrix equations with conjugate of two unknowns

In this work, we are concerned with (R, S) – conjugate solutions to coupled Sylvester complex matrix equations with conjugate of two unknowns. When the considered two matrix equations are consistent, it is demonstrated that the solutions can be obtained by utilizing this iterative algorithm for any initial arbitrary – conjugate matrices . A necessary and sufficient condition is established to guarantee that the proposed method converges to the – conjugate solutions. Finally, two numerical examples are provided to demonstrate the efficiency of the described iterative technique.


Introduction
Many scholars have given complex matrix equations a lot of thought. Chang et al. [1] gives the expression of (R, S) -conjugate solution to the system of matrix equations AX = C, XB = D. Trench [2] investigated the system of linear equations Az = w for R -conjugate matrices; further, as an extension of R -conjugate matrix, Trench [3] defined (R, S) -conjugate matrix and min Az − w for (R, S) -conjugate matrices where z, w are known column vectors. Dong et al. [4] presented the Hermitian R -conjugate solution to the system of complex matrix equations AX = C, XB = D. Ramadan and El-Danaf [5] introduced an iterative method for obtaining the generalized bisymmetric solution to the coupled Sylvester matrix equations (AV + BW, MV + NW) = (EVF + C, GVH + D). Bayoumi and Ramadan [9] presented the Hermitian R -conjugate solution to a generalized coupled Sylvester-conjugate matrix equations. Bayoumi [10] proposed an iterative algorithm for solving a generalized coupled Sylvester -conjugate matrix equations over Hamiltonian matrices. Li et al. [11] presented two algorithms for solving the matrix AXB = C for (R, S)-symmetric matrices X based on the idea of the traditional conjugate gradient method and conjugate gradient Least Squares method.
Trench [12] defined a special class of matrices called (R, S)-symmetric matrices (R, S)-skew symmetric matrices. Balani and Hajarian [13] used the generalized accelerated overrelaxation method and generalized conjugate gradient methods for presenting iteration methods to solve linear systems of equations. Dehghan and Shirilord [14] used the accelerated double-step scale splitting iteration method for solving a class of complex matrix equations. Bayoumi [15] presented two relaxed gradient-based algorithms to find the Hermitian and skew-Hermitian solutions to the linear matrix equation AXB + CXD = F. Ramadan et al. [16] used Sylvester block sum and block matrix Kronecker map to offer an explicit solution of a system of Sylvester matrix equations.
This paper is sorted out as follows: In section 2, we present several notations and lemmas that will play vital roles in the sequel section. In section 3, we provide an iterative algorithm for solving the coupled Sylvester complex matrix equation with conjugate of two unknown over (R, S) -conjugate matrices and we give the convergence properties of these iterative algorithms. In section 4, Two numerical examples are presented to support the theoretical results of the proposed iterative algorithm. denotes the Kronecker product of two matrices A and B. For a matrix X = [x 1 x 2 · · · x n ] ∈ C m×n , vec(X) is the column stretching operation of X, and defined as vec(X) = [x T 1 x T 2 · · · x T n ] T . A well-known feature of the Kronecker product for matrices A, B and C with appropriate dimension, is vec(ABC) = (C T ⊗ A)vec(B). Furthermore, it is obvious that A = vec(A) 2 for any matrix A. Lemma 2.1 [6]: For the matrix equation AXB = F where A ∈ C m×r , B ∈ C s×n and F ∈ C m×n are known matrices and X ∈ C r×s is the matrix to be determined, an iterative algorithm is constructed as If this matrix equation has a unique solution X * , then the iterative solution X(k) converges to the unique solution X * Definition 2.1 [7]: : In the space C m×n over the field R, an inner product space can be described as A, B =

Re[tr(A H B)]
The Frobenius norm of C is denoted by C , that is C = tr(C H C). The matrices B, C ∈ C m×n are called orthogonal if B, C = 0.

Definition 2.2 [3]:
Let R, S be an n × n symmetric orthogonal matrices, that is,

The iterative algorithm
In this section, the following coupled Sylvester complex matrix equations with conjugate of two unknowns are considered (1) where A 11 , A 12 , C 11 , C 12 , A 21 , A 22 , C 21 , C 22 ∈ C m×n , B 11 , B 12 , D 11 , D 12 , B 21 , B 22 , D 21 , D 22 ∈ C n×r and E 1 , E 2 ∈ C m×r are given matrices, while V, W ∈ RSC n×n are matrices to be determined. Denote is consistent.
Proof: If the coupled Sylvester matrix equation (1) has (R, S) -conjugate solutions V * , W * ∈ RSC n×n i.e. RV * S = V * and RW * S = W * , it is obvious that V * , W * are also solutions of equation (2). Conversely assume that the system of matrix equation (2) has solu- Thus, V * , W * are the solutions to the coupled Sylvester matrix equation (1). So the solvability of the coupled Sylvester matrix equation (1) is equivalent to that of matrix equation (2).
Let us rewrite the matrix equation (2) into the equiv- The following is a well-known theorem: We now present the (R, S) -conjugate iterative algorithm to solve the coupled Sylvester matrix equation (1) over (R, S) -conjugate matrices. where 4. If r 1 (k + 1) = 0, r 2 (k + 1) = 0, then stop and V k , W k are the solution; otherwise, put k = k + 1 and go to STEP 3.

Convergence analysis
In this subsection, we present convergence properties of the suggested algorithm I.
These demonstrate that ξ 1 (k), ξ 2 (k) ∈ RSC n×n . Denote Utilizing the aforementioned error matrices and algorithm I, we can obtain Taking the norm of both sides of (6) and (7) and applying the following facts for two square complex matrices tr(AB) = tr(BA), Substituting from the preceding relation into (8), gives Similarly to the preceding, we also write From (9) and (10) Defining the nonnegative definite function η(k) as follows: From the previous results, this function may be calculated as follows: If the convergence factor μ is chosen to satisfy (3), then one has Since the matrix equation (1) has a unique solution pair. It follows from the definition (4) and (5)  This completes the proof of the theorem.

Numerical examples
Two numerical examples are given in this section to test the effectiveness of the algorithms I.
As k increase, the relative error f decreases and eventually disappears, and algorithm I is efficient. Figure 1 depicts the effect of adjusting the convergence factor μ. We can see that the larger the convergence factor μ, the faster the rate of convergence.
There are two approaches for quantifying approximation errors: the residual error and the relative error. The residual error can be misleading as a measure of precision; however the relative error is more useful because the relative error considers the size of the value. Define the residual error as and the relative error as In Table 1, we compare relative error, residual error, and the elapsed CPU time for the convergence factor μ = 3.5 × 10 −4 .

Example 4.2:
Consider the coupled Sylvester complex matrix equations with conjugate of two unknowns With  This coupled Sylvester matrix equation (12) has a unique(R, S) -conjugate solution of the following form.
It can be observed from Figure 2. As k increase, the relative error f decreases and eventually disappears, and algorithm I is effective. We can observe that the larger the convergence factor μ, the faster the convergence rate.

Conclusions
In this paper, we have constructed an effective algorithm to find (R, S) -conjugate solutions to coupled Sylvester complex matrix equations with conjugate of two unknowns. When these two matrix equations are consistent, for any initial arbitrary (R, S) -conjugate matrices V 1 , W 1 the solutions can be obtained by utilizing this iterative algorithm. Sufficient conditions are provided to ensure the proposed algorithm's convergence. We test the proposed algorithm utilizing MATLAB and the results of numerical experiments support our algorithm.

Disclosure statement
No potential conflict of interest was reported by the author(s).