RADI

RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations. This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the Cholesky-factored variant of the Lyapunov ADI method. We discuss important implementation aspects of the algorithm, such as reducing the use of complex arithmetic and shift selection strategies. We show that there is a very tight relation between the new algorithm and three other algorithms for CARE previously known in the literature -- all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution.


References in zbMATH (referenced in 17 articles , 1 standard article )

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  1. Benner, Peter; Heiland, Jan; Werner, Steffen W. R.: Robust output-feedback stabilization for incompressible flows using low-dimensional (\mathcalH_\infty)-controllers (2022)
  2. Hamadi, M. A.; Jbilou, K.; Ratnani, A.: An efficient extended block Arnoldi algorithm for feedback stabilization of incompressible Navier-Stokes flow problems (2022)
  3. Benner, Peter; Köhler, Martin; Saak, Jens: Matrix equations, sparse solvers: \textttM-M.E.S.S.-2.0.1 -- philosophy, features, and application for (parametric) model order reduction (2021)
  4. Benner, Peter; Palitta, Davide: On the solution of the nonsymmetric T-Riccati equation (2021)
  5. Chen, Hao; Wang, Ying: Modified Douglas splitting method for differential matrix equations (2021)
  6. Dorschky, Ines; Reis, Timo; Voigt, Matthias: Balanced truncation model reduction for symmetric second order systems -- a passivity-based approach (2021)
  7. Benner, Peter; Bujanović, Zvonimir; Kürschner, Patrick; Saak, Jens: A numerical comparison of different solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems (2020)
  8. Kressner, Daniel; Kürschner, Patrick; Massei, Stefano: Low-rank updates and divide-and-conquer methods for quadratic matrix equations (2020)
  9. Kürschner, Patrick; Freitag, Melina A.: Inexact methods for the low rank solution to large scale Lyapunov equations (2020)
  10. Liu, Jianzhou; Zhang, Juan; Luo, Fangfang: Newton’s method for the positive solution of the coupled algebraic Riccati equation applied to automatic control (2020)
  11. Zhang, Juan; Kang, Huihui; Tan, Fangyuan: Two-parameters numerical methods of the non-symmetric algebraic Riccati equation (2020)
  12. Zhang, Liping; Fan, Hung-Yuan; Chu, Eric King-wah: Inheritance properties of Krylov subspace methods for continuous-time algebraic Riccati equations (2020)
  13. Breiten, Tobias; Ringh, Emil: Residual-based iterations for the generalized Lyapunov equation (2019)
  14. Jbilou, Khalide; Raydan, Marcos: Nonlinear least-squares approach for large-scale algebraic Riccati equations (2019)
  15. Kürschner, Patrick: Approximate residual-minimizing shift parameters for the low-rank ADI iteration (2019)
  16. Benner, Peter; Bujanović, Zvonimir; Kürschner, Patrick; Saak, Jens: RADI: a low-rank ADI-type algorithm for large scale algebraic Riccati equations (2018)
  17. Målqvist, Axel; Persson, Anna; Stillfjord, Tony: Multiscale differential Riccati equations for linear quadratic regulator problems (2018)