This paper presents an effective low-rank generalized alternating direction implicit iteration (R-GADI) method for solving large-scale sparse and stable Lyapunov matrix equations and continuous-time algebraic Riccati matrix equations. The method is based on generalized alternating direction implicit iteration (GADI), which exploits the low-rank property of matrices and utilizes the Cholesky factorization approach for solving. The advantage of the new algorithm lies in its direct and efficient low-rank formulation, which is a variant of the Cholesky decomposition in the Lyapunov GADI method, saving storage space and making it computationally effective. When solving the continuous-time algebraic Riccati matrix equation, the Riccati equation is first simplified to a Lyapunov equation using the Newton method, and then the R-GADI method is employed for computation. Additionally, we analyze the convergence of the R-GADI method and prove its consistency with the convergence of the GADI method. Finally, the effectiveness of the new algorithm is demonstrated through corresponding numerical experiments.

翻译：本文提出了一种有效的低秩广义交替方向隐式迭代（R-GADI）方法，用于求解大规模稀疏且稳定的Lyapunov矩阵方程和连续时间代数Riccati矩阵方程。该方法基于广义交替方向隐式迭代（GADI），利用矩阵的低秩性质，并采用Cholesky分解技术进行求解。新算法的优势在于其直接且高效的低秩形式，这是Lyapunov GADI方法中Cholesky分解的一种变体，不仅节省了存储空间，还提升了计算效率。在求解连续时间代数Riccati矩阵方程时，首先通过牛顿法将Riccati方程简化为Lyapunov方程，随后利用R-GADI方法进行计算。此外，我们分析了R-GADI方法的收敛性，并证明了其与GADI方法收敛性的一致性。最后，通过相应的数值实验验证了新算法的有效性。