The accelerated gradient based iterative algorithm for solving a class of generalized Sylvester-transpose matrix equation

Abstract

In this paper, we present an accelerated gradient based algorithm by minimizing certain criterion quadratic function for solving the generalized Sylvester-transpose matrix equation $AXB+C{X}^{T}D=F$. The idea is from (Ding and Chen, 2005; Niu et al., 2011; Wang et al., 2012) in which some efficient algorithms were developed for solving the Sylvester matrix equation and the Lyapunov matrix equation. On the basis of the information generated in the previous half-step, we further introduce a relaxation factor to obtain the solution of the generalized Sylvester-transpose matrix equation. We show that the iterative solution converges to the exact solution for any initial value provided that some appropriate assumptions. Finally, some numerical examples are given to illustrate that the introduced iterative algorithm is efficient.

Introduction

In this article, we consider the generalized Sylvester-transpose matrix equation

$$\begin{array}{c}\hfill AXB+C{X}^{T}D=F,\end{array}$$

where $A,B,C,D,F\in {\mathbb{R}}^{n\times n}$ are known matrices and $X\in {\mathbb{R}}^{n\times n}$ is the matrix to be determined.

The linear matrix equations play an important role in some fields of applied mathematics and control theory [1], [7], [8], [9]. Many research works are involved in varieties of systems of matrix equations [4], [5], [6], [10], [11], [12], [13], [15], [17], [18], [19], [20], [22], [24], [25], [26], [28], [32], [33], [35], [36], [38], [40], [41], [42], [43], [44], [45], [47], [48], [49], [50]. Especially, the solvability, solution formula and factorization algorithms about matrix equations are studied in [2], [3], [15], [31], [37], [39] by Bai and his co-authors.

In [23], an iterative algorithm was constructed to solve the equation $AXB=C$. Navarra et al. studied a representation of the general common solution of the matrix equations $A_{1}X{B}_1=C_1,A_{2}X{B}_2=C_2$ [29]. By Moore–Penrose generalized inverse, some necessary and sufficient conditions for the existence of the solution and the expressions of the matrix equation $AX+X^{T}C=B$ are obtained in [34]. In recent years, Dehghan and Hajarian considered the generalized coupled Sylvester matrix equations [14]$AXB+CYD=M,EXF+GYH=N$ and presented a modified conjugate gradient method to solve the generalized coupled Sylvester matrix equations over generalized bisymmetric matrix pair (X, Y). Liang and Liu proposed a modified conjugate gradient method to solve the equation $A_{1}X{B}_1+C_1{X}^T{D}_1=F_1,A_{2}X{B}_2+C_2{X}^T{D}_2=F_2$ [27]. In addition, Deng and Bai studied deeply the consistent conditions and the general expressions about the Hermitian solutions of the linear matrix equation [15]. The closed form solutions to a family of generalized Sylvester matrix equations were given by using the so-called Kronecker matrix polynomials in [46]. In particular, by extending the well-known Jacobi and Gauss–Seidel iterations for $Ax=b,$ Ding et al. gained iterative solutions of the generalized Sylvester matrix equation by the gradient based method [16], [21]. They decompose a system into some subsystems, then the unknown parameters of each subsystem are identified successively. In [31], [37], a relaxed gradient based method and a modified gradient based method are given for solving the Sylvester equation $AX+XB=C,$ respectively.

Inspired by the [21], [31], [37], in this paper, we construct an accelerated gradient based iterative (AGBI) algorithm for solving the generalized Sylvester-transpose matrix equation $AXB+C{X}^{T}D=F$. Theoretical analysis shows that our new method converges to the exact solution for any initial value with some appropriate assumptions. Numerical results illustrate that the method has better convergence performances than those in [21], [31], [37].

As a matter of convenience, we use the following notation throughout this paper: Let ${\mathbb{R}}^{m\times n}$ denotes the set of m × n real matrix. For $A\in {\mathbb{R}}^{m\times n},$ we write A^T, ‖A‖, tr(A) to denote the transpose, the Frobenius norm and trace of the matrix A, respectively. λ(A), σ(A), ρ(A) denote the eigenvalue, singular value and the spectrum radius of the matrix A, respectively. For any $A=\left(a_{ij}\right),B=\left(b_{ij}\right),$ A⊗B denotes the Kronecker product defined as $A\otimes B=\left(a_{ij}B\right),i=1,2,\dots ,m,j=1,2,\dots ,n$. For the matrix $X=(x_1,x_2,\dots ,x_n)\in {\mathbb{R}}^{m\times n},$ vec(X) denotes the vec operator defined as $\mathrm{vec}\left(X\right)={(x_1^T,x_2^T,\dots ,x_n^T)}^T\in {\mathbb{R}}^{mn}$. The inner product in space ${\mathbb{R}}^{m\times n}$ is defined as

$$\begin{array}{c}\hfill 〈A,B〉=\mathrm{tr}\left(B^{T}A\right),\forall A,B\in {\mathbb{R}}^{m\times n},\end{array}$$

particularly, ${\parallel A\parallel }^2=\mathrm{tr}\left(A^{T}A\right)$.

The rest of this paper is organized as follows. In Section 2, we compare our new AGBI algorithm for solving the system (1.1) with the methods in [21], [31], [37]. In Section 3, we made a detailed analysis of the convergence and obtain the optimal convergence factor. Under the appropriate assumptions, we show that the iterative solution converges to the exact solution for any initial value. Some numerical examples are given to illustrate that the introduced iterative algorithm is efficient in Section 4. Conclusions are arranged in Section 5.