In this paper, we first recall some well-known results on the solvability of the generalized Lyapunov equation and rewrite this equation into the generalized Stein equation by using Cayley transformation. Then we introduce the matrix versions of biconjugate residual (BICR), biconjugate gradients stabilized (Bi-CGSTAB), and conjugate residual squared (CRS) algorithms. This study’s primary motivation is to avoid the increase of computational complexity by using the Kronecker product and vectorization operation. Finally, we offer several numerical examples to show the effectiveness of the derived algorithms.

中文翻译：

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求解广义Lyapunov矩阵方程的矩阵迭代算法

在本文中，我们首先回顾一下有关广义Lyapunov方程可解性的一些著名结果，然后通过使用Cayley变换将该方程重写为广义Stein方程。然后，我们介绍双共轭残差（BICR），稳定的双共轭梯度（Bi-CGSTAB）和共轭残差平方（CRS）算法的矩阵版本。这项研究的主要动机是通过使用Kronecker乘积和向量化运算来避免计算复杂性的增加。最后，我们提供了几个数值示例来说明派生算法的有效性。