Purpose

This article proposes a relaxed gradient iterative (RGI) algorithm to solve coupled Sylvester-conjugate transpose matrix equations (CSCTME) with two unknowns.

Design/methodology/approach

This article proposes a RGI algorithm to solve CSCTME with two unknowns.

Findings

The introduced (RGI) algorithm is more efficient than the gradient iterative (GI) algorithm presented in Bayoumi (2014), where the author's method exhibits quick convergence behavior.

Research limitations/implications

The introduced (RGI) algorithm is more efficient than the GI algorithm presented in Bayoumi (2014), where the author's method exhibits quick convergence behavior.

Practical implications

In systems and control, Lyapunov matrix equations, Sylvester matrix equations and other matrix equations are commonly encountered.

Social implications

In systems and control, Lyapunov matrix equations, Sylvester matrix equations and other matrix equations are commonly encountered.

Originality/value

This article proposes a relaxed gradient iterative (RGI) algorithm to solve coupled Sylvester conjugate transpose matrix equations (CSCTME) with two unknowns. For any initial matrices, a sufficient condition is derived to determine whether the proposed algorithm converges to the exact solution. To demonstrate the effectiveness of the suggested method and to compare it with the gradient-based iterative algorithm proposed in [6] numerical examples are provided.

中文翻译：

---

两个未知数耦合西尔维斯特共轭转置矩阵方程的基于松弛梯度的迭代解

目的

本文提出了一种松弛梯度迭代 (RGI) 算法来求解具有两个未知数的耦合西尔维斯特共轭转置矩阵方程 (CSCTME)。

设计/方法论/途径

本文提出了一种 RGI 算法来求解具有两个未知数的 CSCTME。

发现

引入的 (RGI) 算法比 Bayoumi (2014) 中提出的梯度迭代 (GI) 算法更有效，其中作者的方法表现出快速收敛行为。

研究局限性/影响

引入的 (RGI) 算法比 Bayoumi (2014) 中提出的 GI 算法更有效，其中作者的方法表现出快速收敛行为。

实际影响

在系统和控制中，经常遇到Lyapunov矩阵方程、Sylvester矩阵方程和其他矩阵方程。

社会影响

在系统和控制中，经常遇到Lyapunov矩阵方程、Sylvester矩阵方程和其他矩阵方程。

原创性/价值

本文提出了一种松弛梯度迭代 (RGI) 算法来求解具有两个未知数的耦合西尔维斯特共轭转置矩阵方程 (CSCTME)。对于任何初始矩阵，都会导出充分条件来确定所提出的算法是否收敛到精确解。为了证明所提出方法的有效性，并将其与[6]中提出的基于梯度的迭代算法进行比较，提供了数值示例。