Low-rank generalized alternating direction implicit iteration method for solving matrix equations
<|reference_start|>Low-rank generalized alternating direction implicit iteration method for solving matrix equations: 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.<|reference_end|>
@article{zhang2024low-rank,
    title={Low-rank generalized alternating direction implicit iteration method for
  solving matrix equations},
    author={Juan Zhang and Wenlu Xun},
    journal={arXiv preprint arXiv:2404.06034},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.06034},
    primaryClass={math.NA cs.NA}
}