Numerical Algorithms

, Volume 34, Issue 2–4, pp 339–353 | Cite as

Block Krylov Subspace Methods for Large Algebraic Riccati Equations

  • K. Jbilou


In the present paper, we present numerical methods for the computation of approximate solutions to large continuous-time and discrete-time algebraic Riccati equations. The proposed methods are projection methods onto block Krylov subspaces. We use the block Arnoldi process to construct an orthonormal basis of the corresponding block Krylov subspace and then extract low rank approximate solutions. We consider the sequential version of the block Arnoldi algorithm by incorporating a deflation technique which allows us to delete linearly and almost linearly dependent vectors in the block Krylov subspace sequences. We give some theoretical results and present numerical experiments for large problems.

block Arnoldi Krylov subspaces low rank approximations Riccati equations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B.D.O. Anderson and J.B. Moore, Linear Optimal Control (Prentice-Hall, Englewood Cliffs, NJ, 1971).Google Scholar
  2. [2]
    W.F. Arnold, III and A.J. Laub, Generalized eigenproblem algorithms and software for algebraic Riccati equations, Proc. 72 (1984) 1746–1754.Google Scholar
  3. [3]
    Z. Bai and J. Demmel, Using the matrix sign function to compute invariant subspaces, SIAM J. Anal. Appl. 19 (1998) 205–225.Google Scholar
  4. [4]
    P. Benner and R. Byers, An exact line search method for solving generalized continous algebraic Riccati equations, IEEE Trans. Automat. Control 43(1) (1998) 101–107.Google Scholar
  5. [5]
    P. Benner and H. Faβbender, An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem, Linear Algebra Appl. 263 (1997) 75–111.Google Scholar
  6. [6]
    P. Benner, A. Laub and V. Mehrmann, A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case, Technical Report SPC 95-22, Fak. f. Mathematik, TU Chemnitz-Zwickau, Chemnitz, Germany (1995).Google Scholar
  7. [7]
    A. Bunse-Gerstner and V. Mehrmann, A symplectic QR-like algorithm for the solution of the real algebraic Riccati equations, IEEE Trans. Automat. Control 43(1) (1998) 1104–1113.Google Scholar
  8. [8]
    R. Byers, A Hamiltonian QR-algorithm, SIAM J. Sci. Statist. Comput. 7 (1986) 212–229.Google Scholar
  9. [9]
    R. Byers, Solving the algebraic Riccati equation with the matrix sign function, Linear Algebra Appl. 85 (1987) 267–279.Google Scholar
  10. [10]
    J.C. Doyle, K. Glover, P.P. Khargonekar and B.A. Francis, State-space solutions to standar \({\mathcal{H}}_2\) and \({\mathcal{H}}_\infty\) control problems, IEEE Trans. Automat. Control 34(8) (1989) 831–846.Google Scholar
  11. [11]
    E.J. Grimme, D.C. Sorenson and P. Van Dooren, Model reduction of state space systems via an implicitly restarted Lanczos method, Numer. Algorithms 12 (1996) 1–31.Google Scholar
  12. [12]
    J.J. Hench and A.J. Laub, Numerical solution of the discrete-time periodic Riccati equation, IEEE Trans. Automat. Control 39 (1994) 1197–1209.Google Scholar
  13. [13]
    I.M. Jaimoukha and E.M. Kasenally, Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal. 31 (1994) 227–251.Google Scholar
  14. [14]
    C.S. Kenny, A.J. Laub and P. Papadopoulos, Matrix sign function algorithms for Riccati equations, in: Proc. of IMA Conf. on Control: Modelling, Computation, Information, Southend-on-Sea (IEEE Computer Soc. Press, Los Alamitos, CA, 1992) pp. 1–10.Google Scholar
  15. [15]
    D.L. Kleinman, On an iterative technique for Riccati equation computations, IEEE Trans. Automat. Control 13 (1968) 114–115.Google Scholar
  16. [16]
    P. Lancaster and L. Rodman, The Algebraic Riccati Equations (Clarendon Press, Oxford, 1995).Google Scholar
  17. [17]
    A.J. Laub, A Schur method for solving algebraic Riccati equations, IEEE Trans. Automat. Control 24 (1979) 913–921.Google Scholar
  18. [18]
    V. Mehrmann, The Autonomous Linear Quadratic Problem, Theory and Numerical Solution, Lecture Notes in Control and Information Sciences, Vol. 163 (Springer, Heidelberg, 1991).Google Scholar
  19. [19]
    D.J. Roberts, Linear model reduction and solution of the algebraic Riccati equation by use of the sign function, Internat. J. Control 32 (1998) 677–687.Google Scholar
  20. [20]
    A. Ruhe, Rational Krylov sequence methods for eigenvalue computations, Linear Algebra Appl. 58 (1984) 391–405.Google Scholar
  21. [21]
    Y. Saad, Numerical solution of large Lyapunov equations, in: Signal Processing, Scattering, Operator Theory and Numerical Methods, Proc. of the Internat. Symposium MTNS-89, Vol. 3, eds. M.A. Kaashoek, J.H. van Schuppen and A.C. Ran (Birkhäuser, Boston, 1990) pp. 503–511.Google Scholar
  22. [22]
    M. Sadkane, Block Arnoldi and Davidson methods for unsymmetric large eigenvalue problems, Numer. Math. 64 (1993) 687–706.Google Scholar
  23. [23]
    D.C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13(1) (1992) 357–387.Google Scholar
  24. [24]
    G.W. Stewart and J.-G. Sun, Matrix Perturbation Theory (Academic Press, New York, 1990).Google Scholar
  25. [25]
    J.-G. Sun, Perturbation theory for algebraic Riccati equation, SIAM J. Matrix Anal. Appl. 19(1) (1998) 39–65.Google Scholar
  26. [26]
    J.-G. Sun, Residual bounds of approximate solutions of the algebraic Riccati equation, Numer. Math. 76 (1997) 249–263.Google Scholar
  27. [27]
    P. Van Dooren, A generalized eigenvalue approach for solving Riccati equations, SIAM J. Sci. Statist. Comput. 2 (1981) 121–135.Google Scholar
  28. [28]
    C.F. Van Loan, A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix, Linear Algebra Appl. 16 (1984) 233–251.Google Scholar
  29. [29]
    W.M. Wonham, On a matrix Riccati equation of stochastic control, SIAM J. Control 6 (1968) 681–697.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • K. Jbilou
    • 1
  1. 1.Université du Littoral côte d'Opale, Zone universitaire de la Mi-voix, Batiment H. PoincaréCalais CedexFrance

Personalised recommendations