Fast verified computation for solutions of continuous-time algebraic Riccati equations

  • Shinya MiyajimaEmail author
Original Paper Area 2


A fast numerical algorithm for computing interval matrices containing solutions of continuous-time algebraic Riccati equations is proposed. This algorithm utilizes numerical spectral decomposition and involves only cubic complexity. Stabilizing and anti-stabilizing properties and uniqueness of the contained solution can moreover be verified by this algorithm. Numerical results show the property of this algorithm.


Algebraic Riccati equations Stabilizing solution Verified computation 

Mathematics Subject Classification

15A24 65F99 65G20 93C05 



The author thanks Dr. Behnam Hashemi and Prof. Siegfried M. Rump of Shiraz University of Technology and Hamburg University of Technology, respectively, for fruitful discussions. The author also acknowledges the referee for the valuable comments.


  1. 1.
    Abels, J., Benner, P.: CAREX—A collection of benchmark examples for continuous-time algebraic Riccati equations (ver. 2.0). Tech. Rep. SLICOT Working Note 1999-14, (1999)
  2. 2.
    Bavely, A., Stewart, G.: An algorithm for computing reducing subspaces by block diagonalization. SIAM J. Numer. Anal. 16, 359–367 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bini, D.A., Iannazzo, B., Meini, B.: Numerical Solution of Algebraic Riccati Equations. SIAM Publications, Philadelphia (2012)Google Scholar
  4. 4.
    Hashemi, B.: Verified computation of symmetric solutions to continuous-time algebraic Riccati matrix equations. In: Procedings of SCAN conference, Novosibirsk, pp. 54–56 (2012)
  5. 5.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  6. 6.
    Krawczyk, R.: Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken. Computing 4, 187–201 (1969). (in German)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Luther, W., Otten, W.: Verified calculation of the solution of algebraic Riccati equation. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 105–118. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  8. 8.
    Luther, W., Otten, W., Traczinski, H.: Verified calculation of the solution of continuous- and discrete time algebraic Riccati equation. Schriftenreihe des Fachbereichs Mathematik der Gerhard-Mercator-Universität Duisburg, Duisburg, SM-DU-422 (1998)Google Scholar
  9. 9.
    Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM Publications, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  10. 10.
    Minamihata, A.: private communication (2013)Google Scholar
  11. 11.
    Miyajima, S.: Fast enclosure for solutions of Sylvester equations. Linear Algebr. Appl. 439, 856–878 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Miyajima, S.: Fast enclosure for all eigenvalues and invariant subspaces in generalized eigenvalue problems. SIAM J. Matrix Anal. Appl. 35, 1205–1225 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rump, S.M.: INTLAB—INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–107. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  14. 14.
    Rump, S.M.: INTLAB—INTerval LABoratory, the MATLAB toolbox for verified computations (ver. 5.3). (2006)
  15. 15.
    Yano, K., Koga, M.: Verified numerical computation in LQ control problem. Trans. SICE 45, 261–267 (2009). (in Japanese)CrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan 2015

Authors and Affiliations

  1. 1.Faculty of EngineeringGifu UniversityGifuJapan

Personalised recommendations