Основні поняття
This paper presents an effective low-rank generalized alternating direction implicit iteration (R-GADI) method for efficiently solving large-scale sparse and stable Lyapunov matrix equations and continuous-time algebraic Riccati matrix equations. The method exploits the low-rank property of matrices and utilizes Cholesky factorization, providing a direct and efficient low-rank formulation that saves storage space and computational cost.
Анотація
The paper focuses on developing an efficient numerical solution for large-scale continuous-time algebraic Riccati matrix equations (CARE) and Lyapunov matrix equations.
Key highlights:
- The R-GADI method is proposed, which is an improvement over the GADI algorithm for solving the Lyapunov equation. It represents the solution as a low-rank approximation, eliminating the need to store the full matrix and reducing storage requirements.
- The Kleinman-Newton method is combined with R-GADI (Kleinman-Newton-RGADI) to solve the Riccati equation, significantly reducing the total number of ADI iterations and lowering the overall computational cost.
- Convergence analysis is provided, proving the consistency between R-GADI and GADI iterations.
- Numerical experiments demonstrate the effectiveness of the proposed R-GADI method in solving large-scale Lyapunov and Riccati matrix equations compared to other existing methods.
Статистика
The paper presents numerical results for solving Lyapunov and Riccati matrix equations of varying dimensions. Some key data points include:
For the Lyapunov equation with n=4096, the R-GADI method with α=max σ(F) achieved a relative residual of 8.887e-16 in 7 iterations, taking 1223.03 seconds.
For the Riccati equation with n=4096, the R-GADI method achieved a relative residual of 2.983e-16 in 9 iterations, taking 1495.60 seconds, outperforming other methods like GADI, R1-ADI, and R2-ADI.