  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.
