An accelerated gradient-based iterative algorithm for solving extended Sylvester

Abstract

In this paper, we present an accelerated gradient-based iterative algorithm for solving extended Sylvester–conjugate matrix equations. The idea is from the gradient-based method introduced in Wu et al. (Applied Mathematics and Computation 217(1): 130–142, 2010a) and the relaxed gradient-based algorithm proposed in Ramadan et al. (Asian Journal of Control 16(5): 1–8, 2014) and the modified gradient-based algorithm proposed in Bayoumi (PhD thesis, Ain Shams University, 2014). The convergence analysis of the algorithm is investigated. We show that the iterative solution converges to the exact solution for any initial value provided some appropriate assumptions be made. A numerical example is given to illustrate the effectiveness of the proposed method and to test its efficiency and accuracy compared with an existing one presented in Wu et al. (2010a), Ramadan et al. (2014) and Bayoumi (2014).

Keywords

Convergence extended Sylvester–conjugate matrix equations Frobenius norm gradient-based algorithm modified gradient relaxation parameters

Introduction

Linear matrix equations play an important role in some fields of applied mathematics and control theory (Bevis et al., 1987; Huang et al., 2008; Ramadan et al., 2014; Xie et al., 2010). The gradient-based iterative algorithm in Ding and Chen (2005a, 2005b, 2006) and least squares-based iterative algorithm (Ding and Chen, 2006) for solving (coupled) matrix equations are efficient numerical algorithms presented based on the hierarchical identification principle (Ding and Chen, 2005a, 2005c). In Xie et al. (2010), gradient-based and a least squares-based iterative algorithms for solving matrix equation $AXB + C X^{T} D = F$ are presented. Ding and Zhang (2014a) proposed a gradient-based iteration established for solving the coupled matrix equations $A_{i} X B_{i} = F$ . In Ding and Zhang (2016a), by using the hierarchical identification principle and introducing the convergence factor and the iterative matrix, a family of inversion-free iterative algorithms is proposed for solving non-linear matrix equations $X + A^{T} X^{- 1} A = I$ .

In the matrix algebra field, some complex matrix equations have attracted much attention from many researchers, as it was shown in Bevis et al. (1987) that the consistence of the matrix equation $AX - \bar{X} B = C$ was related to the consimilarity (Horn and Johnson, 1990; Huang, 2001; Jiang et al., 2006) of two partitioned matrices associated with the matrices $A; B$ and C. Huang et al. (2008) proposed an iterative method for solving the linear matrix equation $AXB = F$ over skew-symmetric matrix X. Ding et al. (2008) derived iterative solutions of matrix equation $AXB = F$ and the Sylvester matrix equation $AXB + CXD = F$ .

Recently, Niu et al. (2011) proposed a relaxed gradient-based iterative algorithm (RGI) for solving Sylvester equations $AX + XB = C$ . The numerical experiments show that the convergent behaviour of Niu’s algorithm is better than Ding’s algorithm (Ding and Chen, 2005b). Wang et al. (2012) proposed a modified gradient-based algorithm (MGI) for solving Sylvester equation $AX + XB = C$ .

Iterative algorithms for solving complex matrix equation have also attracted much attention. By applying the hierarchical identification principle, an iterative algorithm was constructed in Wu et al. (2010a) to solve the so-called extended Sylvester–conjugate matrix equations. Ramadan et al. (2014) proposed an RGI for solving extended Sylvester–conjugate matrix equations $AXB + C \bar{X} D = F$ . Bayoumi (2014) proposed an MGI for solving extended Sylvester–conjugate matrix equations. In Wu et al. (2010c), the matrix equation $X - A \bar{X} B = C$ is considered, which included the Sylvester–conjugate and Yakubovich–conjugate matrix equations as special cases. In Wu et al. (2010b), a finite iterative algorithm was proposed to solve more general complex matrix equations. Ding and Zhang (2014b) discussed the properties of the eigenvalues related to the symmetric positive definite matrices. The iterative and recursive algorithms have many important applications in science and engineering areas, such as signal processing and system identification (Ding et al., 2016b, 2016c) and filtering (Ding et al., 2016d; Wang and Ding, 2016). Zhou et al. (2008, 2009) was concerned with iterative methods for solving a class of coupled matrix equations, including the well-known coupled Markovian jump Lyapunov matrix equations as special cases.

This paper is organized as follows: first, in the next section, we introduce some notations and lemmas that will be needed to develop this work. Then, we introduce brief reviews of the gradient-based iterative algorithm proposed in Wu et al. (2010a), the RGI for solving the problem under consideration in Ramadan et al. (2014) and the MGI for solving the problem under consideration in Bayoumi (2014). We propose an iterative algorithm to obtain the solutions to the extended Sylvester–conjugate matrix equations by using an accelerated gradient-based iterative algorithm (AGI) and give the convergence properties of this iterative algorithm. Finally, a numerical example is given to explore the effectiveness, efficiency and accuracy of the presented method.

Preliminaries

The following notations, definitions and lemmas will be used to develop the proposed work. We use $A^{T}, \bar{A}, A^{H}$ , $‖ A ‖_{2}$ and $tr (A)$ to denote the transpose, conjugate, conjugate transpose, spectral norm of A and the trace of A, respectively. We denote the set of all $m \times n$ complex matrices by $ℂ^{m \times n}$ . The Frobenius norm of the matrix A is denoted by $‖ A ‖ = \sqrt{{〈 A, A 〉}_{r}} = \sqrt{Re (tr (A^{H} A))}$ .

Lemma 1 (Zhou et al., 2009). Consider the following matrix equation

AXB = F

where $A \in$ $ℂ^{m \times r}$ , $B \in$ $ℂ^{s \times n}$ and $F \in$ $ℂ^{m \times n}$ are known matrices, and $X \in$ $ℂ^{r \times s}$ is the matrix to be determined. For this matrix equation, an iterative algorithm is constructed as

X (k + 1) = X (k) + μ A^{H} (F - AX (k) B) B^{H}

with

0 < μ < \frac{2}{‖ A ‖_{2}^{2} ‖ B ‖_{2}^{2}} .

If this matrix equation has a unique solution X, then the iterative solution $X (k)$ converges to the unique solution X, that is, $lim_{k \to \infty} X (k) = X$ .

Lemma 2 (Von Neumann, 1937). Suppose M and N are two $n \times n$ complex matrices with their singular values

$ρ_{1} \geq \dots \geq ρ_{n}$ ; $σ_{1} \geq \dots \geq σ_{n}$ respectively, then we have $| trace (MN) | \leq \sum_{i = 1}^{n} ρ_{i} σ_{i}$ .

Remark 1. If N is symmetric positive definite, then $trace (N) = \sum_{i = 1}^{n} λ_{i} (N) = \sum_{i = 1}^{n} σ_{i} (N)$ . Therefore, in this case we have $| trace (MN) | \leq ρ_{1} \sum_{i = 1}^{n} σ_{i} = ρ_{1} trace (N)$ .

Brief review of the gradient-based iterative algorithm (GI), the relaxed gradient-based iterative algorithm (RGI) and modified gradient-based iterative algorithm (MGI)

Wu et al. (2010a) proposed the following gradient-based algorithm (GI) for solving the equation $AXB + C \bar{X} D = F$ .

Algorithm 1 (The gradient-based iterative (GI) algorithm)

By regarding the extended Sylvester–conjugate matrix equation

AXB + C \bar{X} D = F

(1)

as two linear matrix equations of the form

AXB = F - C \bar{X} D

(2)

\bar{C} X \bar{D} = \bar{F} - \bar{AXB}

(3)

Wu et al. (2010a) presented the following algorithm for solving Equation (1)

Algorithm 1

Given any two initial approximate solution block vectors $X_{1} (0)$ , $X_{2} (0)$ and then $X (0) = [X_{1} (0) + X_{2} (0)] / 2$ .

For $k = 1, 2, \dots$ until converges

\begin{matrix} X_{1} (k) = X (k - 1) + μ A^{H} (F - AX (k - 1) B - C \bar{X (k - 1)} D) B^{H} \end{matrix}

(4)

X_{2} (k) = X (k - 1) + μ {\bar{C}}^{H} (\bar{F} - \bar{AX (k - 1) B} - \bar{C} X (k - 1) \bar{D}) {\bar{D}}^{H}

(5)

X (k) = \frac{X_{1} (k) + X_{2} (k)}{2} .

(6)

End

It is shown in Wu et al. (2010a) that the gradient-based iterative algorithm converges as long as

0 < μ < \frac{1}{‖ A ‖_{2}^{2} ‖ B ‖_{2}^{2} + ‖ C ‖_{2}^{2} ‖ D ‖_{2}^{2}} .

Ramadan et al. (2014) presented the following algorithm for solving Equation (1).

Algorithm 2 (The relaxed gradient-based iterative (RGI) algorithm)

Given any two initial approximate solution block vectors $X_{1} (0)$ , $X_{2} (0)$ and $X (0) = [X_{1} (0) + X_{2} (0)] / 2$ .

For $k = 1, 2, \dots$ until converges

\begin{matrix} X_{1} (k) = X_{1} (k - 1) + μ (1 - ω) A^{H} \\ (F - AX (k - 1) B - C \bar{X (k - 1)} D) B^{H} . \end{matrix}

(7)

\begin{matrix} X_{2} (k) = X_{2} (k - 1) + μ ω {\bar{C}}^{H} \\ (\bar{F} - \bar{AX (k - 1) B} - \bar{C} X (k - 1) \bar{D}) {\bar{D}}^{H} . \end{matrix}

(8)

X (k) = ω X_{1} (k) + (1 - ω) X_{2} (k) .

(9)

End

where $ω$ is a relaxation parameter satisfying $0 < ω < 1$ ; it controls the relative importance of two residual matrices.

It is shown in Ramadan et al. (2014) that the relaxed gradient iterative algorithm converges as long as

0 < μ < \frac{2}{ω (1 - ω) (λ_{\max} (A A^{H}) λ_{\max} (B^{H} B) + λ_{\max} (\bar{C} {\bar{C}}^{H}) λ_{\max} ({\bar{D}}^{H} \bar{D}) + 2 σ_{\max} (\bar{D} \bar{C} A^{H} B^{H}))}

Bayoumi (2014) presented the following algorithm for solving Equation (1).

Algorithm 3 (The MGI algorithm)

Given any two initial approximate solution block vectors $X_{1} (0)$ , $X_{2} (0)$ and then $X (0) = [X_{1} (0) + X_{2} (0)] / 2$ .

For $k = 1, 2, \dots$ until converges

$X_{1} (k) = X (k - 1) + μ A^{H} [F - AX (k - 1) B - C \bar{X (k - 1)} D] B^{H}$ .

$X (k - 1) = [X_{1} (k) + X_{2} (k - 1)] / 2$ .

$X_{2} (k) = X (k - 1) + μ {\bar{C}}^{H} [\bar{F} - \bar{AX (k - 1) B} - \bar{C} X (k - 1) \bar{D}] {\bar{D}}^{H}$ .

$X (k) = [X_{1} (k) + X_{2} (k)] / 2$ .

End

One must note that in the step of computing $X_{2} (k)$ , the last approximate solution $X_{1} (k)$ has been computed. Hence, we can use the information of $X_{1} (k)$ to update the $X (k - 1)$ .

It is shown in Bayoumi (2014) that the modified gradient iterative algorithm converges as long as

0 < μ < min {\frac{2}{λ_{max} (A A^{H}) λ_{max} (B^{H} B)}, \frac{2}{λ_{max} (\bar{C} {\bar{C}}^{H}) λ_{max} ({\bar{D}}^{H} \bar{D})}}

Next, we will compare the AGI algorithm with the existing methods.

The main results

Iterative algorithm

In this subsection, an iterative solution to the extended Sylvester–conjugate matrix equation $AXB + C \bar{X} D = F$ , given in (1), where $A, C \in$ $ℂ^{m \times r}$ , $B, D \in$ $ℂ^{s \times n}$ and $F \in$ $ℂ^{m \times n}$ are given matrices, while $X \in$ $ℂ^{r \times s}$ , is the matrix to be determined. Now we are in the position to present the following algorithm:

Algorithm 4

Step 1: Input matrices $A, C \in$ $ℂ^{m \times r}$ , $B, D \in$ $ℂ^{s \times n}$ and $F \in$ $ℂ^{m \times n}$ and numbers $ε$ and $ω$ . Choose initial matrices $X_{1} (0)$ , $X_{2} (0)$ . Compute $X (0) = ω X_{1} (0) + (1 - ω) X_{2} (0)$ . Set $k : = 1$ .

Step 2: If $δ_{k} = ‖ AX (k) B + C \bar{X (k)} D - F ‖ / ‖ F ‖ < ε$ , stop; otherwise go to step 3.

Step 3: Update the sequence

\begin{matrix} X_{1} (k) = X (k - 1) + μ (1 - ω) A^{H} (F - AX (k - 1) B \\ - C \bar{X (k - 1)} D) B^{H} \end{matrix}

(10)

Step 4: Compute

X (k - 1) = ω X_{1} (k) + (1 - ω) X_{2} (k - 1)

(11)

Step 5: Update the sequence

\begin{matrix} X_{2} (k) = X (k - 1) + μ ω {\bar{C}}^{H} (\bar{F} - \bar{AX (k - 1) B} \\ - \bar{C} X (k - 1) \bar{D}) {\bar{D}}^{H} \end{matrix}

(12)

Step 6: Compute

X (k) = ω X_{1} (k) + (1 - ω) X_{2} (k)

(13)

Step 7: Set $k = k + 1$ and return to step 2

Step 8: End where v is a relaxation parameter satisfying 0 v 1; it controls the relative importance of two residual matrices.

One must note that in the step of computing $X_{2} (k)$ , the last approximate solution $X_{1} (k)$ has been computed. Hence, we can use the information of $X_{1} (k)$ to update the $X (k - 1)$ .

Convergence analysis

In this subsection, a theorem is stated and proved to investigate the convergence properties of the proposed algorithm.

Theorem. If the matrix equation (1) is consistent and has a unique solution $X^{*}$ and

\begin{matrix} 0 < μ < min \\ {\frac{2}{(1 - ω) λ_{max} (A A^{H}) λ_{max} (B^{H} B)}, \frac{2}{ω λ_{max} (\bar{C} {\bar{C}}^{H}) λ_{max} ({\bar{D}}^{H} \bar{D})}} \end{matrix}

(14)

Then the iterative sequence ${X (k)}$ generated by algorithm 4 converges to $X^{*}$ that is $lim_{k \to \infty} X (k) = X^{*}$ ; or the error $\tilde{X} (k) = X (k) - X^{*}$ converges to zero for any initial matrix $X (0)$ .

Proof. First we define the estimation error matrices as

\begin{matrix} \tilde{X} (k) = X (k) - X^{*}, {\tilde{X}}_{1} (k) = X_{1} (k) - X^{*}, \\ {\tilde{X}}_{2} (k) = X_{2} (k) - X^{*} and \tilde{\hat{X}} = \tilde{X} (k) - X^{*} \end{matrix}

(15)

for $k = 1, 2, \dots$ .

and

\begin{matrix} ε (k) = A \tilde{X} (k - 1) B, η (k) = \bar{C} \tilde{X} (k - 1) \bar{D}, \\ δ (k) = A \tilde{\hat{X}} (k - 1) B and ξ (k) = \bar{C} \tilde{\hat{X}} (k - 1) \bar{D} \end{matrix}

(16)

By using the above error matrices (15), (16) and our algorithm and the matrix equation (1) it is easy to get

\begin{matrix} {\tilde{X}}_{1} (k) = \tilde{X} (k - 1) + μ (1 - ω) A^{H} \\ (F - AX (k - 1) B - C \bar{X (k - 1)} D) B^{H} \\ {\tilde{X}}_{1} (k) = \tilde{X} (k - 1) + μ (1 - ω) A^{H} \\ (A X^{*} B + C \bar{X^{*}} D - AX (k - 1) B - C \bar{X (k - 1)} D) B^{H} \\ {\tilde{X}}_{1} (k) = \tilde{X} (k - 1) - μ (1 - ω) A^{H} \\ (A \tilde{X} (k - 1) B + C \bar{\tilde{X} (k - 1)} D) B^{H} \\ {\tilde{X}}_{1} (k) = \tilde{X} (k - 1) + μ (1 - ω) A^{H} (- ε (k) - \bar{η (k)}) B^{H} \end{matrix}

(17)

Similarly to the above, we can write

{\tilde{X}}_{2} (k) = \tilde{\hat{X}} (k - 1) + μ ω {\bar{C}}^{H} (\bar{- δ (k)} - ξ (k)) {\bar{D}}^{H}

(18)

Now, by taking the Frobenius norm of both sides of (17) and (18), it follows that

\begin{matrix} ‖ {\tilde{X}}_{1} (k) ‖^{2} = tr ({\tilde{X}}_{1}^{H} (k) {\tilde{X}}_{1} (k)) \\ = ‖ \tilde{X} (k - 1) ‖^{2} + 2 μ (1 - ω) tr (B^{H} {\tilde{X}}^{H} (k - 1) \\ A^{H} (- ε (k) - \bar{η (k)})) \\ + μ^{2} (1 - ω)^{2} [‖ A^{H} (- ε (k) - \bar{η (k)}) B^{H} ‖^{2}] \\ \leq ‖ \tilde{X} (k - 1) ‖^{2} + 2 μ (1 - ω) tr (ε^{H} (k) \\ (- ε (k) - \bar{η (k)})) + μ^{2} (1 - ω)^{2} [‖ A^{H} (- ε (k) - \bar{η (k)}) B^{H} ‖^{2}] \\ \leq ‖ \tilde{X} (k - 1) ‖^{2} + 2 μ (1 - ω) tr (ε^{H} (k) (- ε (k) - \bar{η (k)})) \\ + μ^{2} (1 - ω)^{2} λ_{max} (A A^{H}) λ_{max} (B^{H} B) ‖ ε (k) + \bar{η (k)} ‖^{2} \end{matrix}

Similarly

$‖ {\tilde{X}}_{2} (k) ‖^{2} \leq ‖ \tilde{\hat{X}} (k - 1) ‖^{2} + 2 μ ω tr (ξ^{H} (k) (\bar{- δ (k)} - ξ (k))) + μ^{2} ω^{2} λ_{max} (\bar{C} {\bar{C}}^{H}) λ_{max} ({\bar{D}}^{H} \bar{D}) ‖ \bar{δ (k)} + ξ (k) ‖^{2}$

According to $X (k) = ω X_{1} (k) + (1 - ω) X_{2} (k)$ we have

\tilde{X} (k) = ω {\tilde{X}}_{1} (k) + (1 - ω) {\tilde{X}}_{2} (k)

\begin{matrix} ‖ \tilde{X} (k) ‖^{2} = ‖ ω {\tilde{X}}_{1} (k) + (1 - ω) {\tilde{X}}_{2} (k) ‖^{2} \\ \leq ω^{2} ‖ {\tilde{X}}_{1} (k) ‖^{2} + (1 - ω)^{2} ‖ {\tilde{X}}_{2} (k) ‖^{2} \\ \leq ω^{2} ‖ \tilde{X} (k - 1) ‖^{2} + 2 μ ω^{2} \\ (1 - ω) tr (ε^{H} (k) (- ε (k) - \bar{η (k)})) \\ + μ^{2} ω^{2} (1 - ω)^{2} λ_{max} (A A^{H}) λ_{max} (B^{H} B) ‖ ε (k) + \bar{η (k)} ‖^{2} \\ + (1 - ω)^{2} ‖ \tilde{\hat{X}} (k - 1) ‖^{2} \\ + 2 μ ω (1 - ω)^{2} tr (ξ^{H} (k) (\bar{- δ (k)} - ξ (k))) \\ + μ^{2} ω^{2} (1 - ω)^{2} λ_{max} (\bar{C} {\bar{C}}^{H}) λ_{max} ({\bar{D}}^{H} \bar{D}) ‖ \bar{δ (k)} + ξ (k) ‖^{2} \end{matrix}

\begin{matrix} \leq ω^{2} ‖ \tilde{X} (k - 1) ‖^{2} + (1 - ω)^{2} ‖ \tilde{\hat{X}} (k - 1) ‖^{2} \\ - ω^{2} [2 μ (1 - ω) - μ^{2} (1 - ω)^{2} λ_{max} (A A^{H}) \\ λ_{max} (B^{H} B)] ‖ ε (k) + \bar{η (k)} ‖^{2} \\ - (1 - ω)^{2} [2 μ ω - μ^{2} ω^{2} λ_{max} \\ (\bar{C} {\bar{C}}^{H}) λ_{max} ({\bar{D}}^{H} \bar{D})] ‖ \bar{δ (k)} + ξ (k) ‖^{2} \\ \leq ω^{2} ‖ \tilde{X} (0) ‖^{2} + (1 - ω)^{2} \sum_{s = 1}^{k} {‖ \tilde{\hat{X}} (k - s) ‖}^{2} \\ - ω^{2} [2 μ (1 - ω) - μ^{2} (1 - ω)^{2} λ_{max} (A A^{H}) λ_{max} (B^{H} B)] \\ \sum_{s = 1}^{k} {‖ ε (s) + \bar{η (s)} ‖}^{2} \\ - (1 - ω)^{2} [2 μ ω - μ^{2} ω^{2} λ_{max} (\bar{C} {\bar{C}}^{H}) λ_{max} ({\bar{D}}^{H} \bar{D})] \\ \sum_{s = 1}^{k} {‖ \bar{δ (s)} + ξ (s) ‖}^{2} \end{matrix}

Thus, $\sum_{s = 1}^{k} {‖ \tilde{\hat{X}} (k - s) ‖}^{2} < \infty$ . For the necessary condition of the series convergence, we have $lim_{k \to \infty} \tilde{\hat{X}} (k) = 0$ when the convergence factor $μ$ satisfies

\begin{array}{l} 0 < μ < \frac{2}{(1 - ω) λ_{\max} (A A^{H}) λ_{\max} (B^{H} B)}, \\ and 0 < μ < \frac{2}{ω λ_{\max} (\bar{C} {\bar{C}}^{H}) λ_{\max} ({\bar{D}}^{H} \bar{D})} \end{array}

Namely

\begin{matrix} 0 < μ < min \\ {\frac{2}{(1 - ω) λ_{max} (A A^{H}) λ_{max} (B^{H} B)}, \frac{2}{ω λ_{max} (\bar{C} {\bar{C}}^{H}) λ_{max} ({\bar{D}}^{H} \bar{D})}} \end{matrix}

we have

\begin{matrix} \sum_{s = 1}^{k} {‖ ε (s) + \bar{η (s)} ‖}^{2} < \infty \\ \sum_{s = 1}^{k} {‖ \bar{δ (s)} + ξ (s) ‖}^{2} < \infty \end{matrix}

It follows from the convergence of the series that

lim_{k \to \infty} \sum_{s = 1}^{k} {‖ ε (s) + \bar{η (s)} ‖}^{2} = 0

and

lim_{k \to \infty} \sum_{s = 1}^{k} {‖ \bar{δ (s)} + ξ (s) ‖}^{2} = 0

This implies that

lim_{k \to \infty} ε (k) + \bar{η (k)} = 0

and

lim_{k \to \infty} \bar{δ (k)} + ξ (k) = 0

i.e. $A \tilde{X} (k) B + C \bar{\tilde{X} (k)} D \to 0$ , which give $\tilde{X} (k) \to 0$ as $k \to \infty$ and we get $lim_{k \to \infty} X (k) = X^{*}$ .

This completes the proof of the theorem.

Numerical example

In the following tests, the convergence factor $μ$ is set to be $\frac{2}{‖ A ‖_{2}^{2} ‖ B ‖_{2}^{2} + ‖ C ‖_{2}^{2} ‖ D ‖_{2}^{2}}$ in the GI (Wu et al., 2010a), and to be

\frac{2}{ω (1 - ω) (λ_{max} (A A^{H}) λ_{max} (B^{H} B) + λ_{max} (\bar{C} {\bar{C}}^{H}) λ_{max} ({\bar{D}}^{H} \bar{D}) + 2 σ_{max} (\bar{D} \bar{C} A^{H} B^{H}))}

where $σ_{max}$ denotes the largest singular value of the matrix $\bar{D} \bar{C} A^{H} B^{H}$ in the RGI (Ramadan et al., 2014). Meanwhile, the relaxation factor $ω$ is set to be 0.8 in the RGI, and to be

min {\frac{2}{λ_{max} (A A^{H}) λ_{max} (B^{H} B)}, \frac{2}{λ_{max} (\bar{C} {\bar{C}}^{H}) λ_{max} ({\bar{D}}^{H} \bar{D})}}

in the MGI (Bayoumi, 2014) and to be

0 < μ < min {\frac{2}{(1 - ω) λ_{max} (A A^{H}) λ_{max} (B^{H} B)}, \frac{2}{ω λ_{max} (\bar{C} {\bar{C}}^{H}) λ_{max} ({\bar{D}}^{H} \bar{D})}}

in the AGI. Meanwhile, the relaxation factor $ω$ is set to be 0.8. The residual norms are recorded and plotted by MATLAB command semilogy.

We consider the following example, where we illustrate the theoretical results of our algorithm for the extended Sylvester–conjugate matrix equation $AXB + C \bar{X} D = F$ .

Given

\begin{matrix} A = [\begin{matrix} 1 + 2 i & 2 - i \\ 1 - i & 2 + 3 i \end{matrix}], B = [\begin{matrix} 2 - 4 i & i \\ - 1 + 3 i & 2 \end{matrix}], \\ C = [\begin{matrix} - 1 - i & - 3 i \\ 0 & 1 + 2 i \end{matrix}], \\ D = [\begin{matrix} - 2 & 1 - i \\ 1 + i & - 1 - i \end{matrix}], F = [\begin{matrix} 21 + 11 i & - 9 + 7 i \\ 52 - 22 i & - 18 + i \end{matrix}] \end{matrix}

while the initial matrix is chosen as

X (0) = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]

This matrix equation has unique solution

X = [\begin{matrix} 1 + 2 i & - i \\ 2 + i & - 1 + i \end{matrix}] .

The coefficient matrices used in this example are taken from Wu et al. (2010a). In Figure 1, the convergence curves by the four iterative solvers are recorded. From the figure, we can see that the proposed AGI converges faster than the MGI, RGI and GI.

Figure 1.

Comparison of convergence curves.

According to corollary 1 in Wu et al. (2010a), the algorithm in Wu et al. (2010a) is convergent for $0 < μ < 0.0028$ . According to theorem 1 in Ramadan et al. (2014), the algorithm in Ramadan et al. (2014) is convergent for $0 < μ < 0.0074$ . According to theorem 1 in Bayoumi (2014), the algorithm in Bayoumi (2014) is convergent for $0 < μ < 0.0035$ . According to the theorem in our work, the proposed algorithm is convergent for $0 < μ < 0.0170$ . We can see in Figure 1 that the iteration stops at $k = 54$ in our case (AGI), while the iteration stops at $k = 82$ in Bayoumi (2014) (MGI), stops at $k = 141$ in Wu et al. (2010a) and stops at $k = 110$ in Ramadan et al. (2014) (RGI). Define the relative iterative error as

f (k) = \frac{‖ X (k) - X^{*} ‖}{‖ X^{*} ‖}

From Figure 1, it is clear that the error $f (k)$ becomes smaller and tends to zero as k increases. The iterative solution $X (k)$ for (AGI) is given in Table 1 for $μ = 0.0170$ .

Table 1.

The iterative solution for the accelerated gradient-based iterative algorithm (AGI) for $μ = 0.0170$ .

k	$x_{11}$	$x_{12}$	$x_{21}$	$x_{22}$
10	1.0702+1.8798i	0.0004−1.1497i	1.8710 +0.9919i	−1.1795+0.9254i
20	1.0517+1.9249i	0.0522−1.0471i	1.9253+0.9853i	−1.0776+0.9543i
30	1.0345+1.9648i	0.0424−1.0360i	1.9777+0.9837i	−1.0320+0.9862i
40	1.0210+1.9752i	0.0261−1.0231i	1.9885+0.9942i	−1.0140+0.9893i
50	1.0117+1.9826i	0.0161−1.0173i	1.9965+0.9959i	−1.0056+0.9953i
54	1.0094+1.9846i	0.0131−1.0155i	1.9980+0.9972i	−1.0035+0.9962i
Solution	$1 + 2 i$	$- i$	$2 + i$	$- 1 + i$

In Table 2, we introduce comparison between our proposed AGI, the MGI (Bayoumi, 2014), the RGI (Ramadan et al., 2014) and the GI (Wu et al., 2010a) for different residuals

Table 2.

Residual versus the number of iteration k for accelerated gradient-based iterative algorithm (AGI), modified gradient-based iterative algorithm (MGI), relaxed gradient-based iterative algorithm (RGI) and gradient-based iterative algorithm (RGI) (GI).

Residual $r_{k}$	The number of iterations: k
Residual $r_{k}$	AGI	MGI	RGI	GI
0.1	54	82	110	141
0.01	138	198	251	338
0.001	271	387	486	655

We can see in Figure 2 and Table 3 the effect of changing the convergence factor $μ$ . We can see that the larger the convergence factor $μ$ , the faster the convergence rate.

Figure 2.

The convergence performance for different convergence factor $μ$ .

Table 3.

The convergence factor $μ$ versus the number of iteration k.

Convergence factor $μ$	The number of iteration k
0.0170	54
0.0120	79
0.0070	140

We can see in Figure 3 and Table 4 the effect of changing the relaxation parameter $ω$ . We can see that the larger the relaxation parameter $ω$ , the faster the convergence rate.

Figure 3.

The convergence performance for different relaxation parameter $ω$ .

Table 4.

The relaxation parameter $ω$ versus the number of iterations k.

Relaxation parameter $ω$	The number of iteration k
0.6	67
0.7	58
0.8	54

Conclusions

In this paper, we have introduced an AGI for solving extended Sylvester–conjugate matrix equations $AXB + C \bar{X} D = F$ . We show that the iterative solution converges to the exact solution for any initial value provided that some sufficient convergence condition, and numerical experiments are performed to illustrate the performance of the proposed algorithm using MATLAB. The proposed method is compared with the existing ones presented in Wu et al. (2010a), Ramadan et al. (2014) and Bayoumi (2014), where our method exhibits fast convergence behaviour.

Footnotes

Conflict of Interest Statement

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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An accelerated gradient-based iterative algorithm for solving extended Sylvester–conjugate matrix equations

Abstract

Keywords

Introduction

Preliminaries

Brief review of the gradient-based iterative algorithm (GI), the relaxed gradient-based iterative algorithm (RGI) and modified gradient-based iterative algorithm (MGI)

Algorithm 1 (The gradient-based iterative (GI) algorithm)

Algorithm 1

Algorithm 2 (The relaxed gradient-based iterative (RGI) algorithm)

Algorithm 3 (The MGI algorithm)

The main results

Iterative algorithm

Algorithm 4

Convergence analysis

Numerical example

Conclusions

Footnotes

Conflict of Interest Statement

Funding

References