Skip to main content
Log in

Normwise, mixed and componentwise condition numbers of nonsymmetric algebraic Riccati equations

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

This paper is devoted to the perturbation analysis for nonsymmetric algebraic Riccati equations. The upper bounds for the normwise, mixed and componentwise condition numbers are presented. The results are illustrated by numerical examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Bao, L., Lin, Y., Wei, Y.: A modified simple iterative method for nonsymmetric algebraic Riccati equations arising in transport theory. Appl. Math. Comput. 181, 1499–1504 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Clancey, K., Gohberg, I.: Factorization of matrix functions and singular integral operators. In: Operator Theory: Advances and Applications, vol. 3. Birkhäuser, Basel (1981)

    Google Scholar 

  3. Cucker, F., Diao, H., Wei, Y.: On mixed and componentwise condition numbers for Moore-Penrose inverse and linear least squares problems. Math. Comput. 76, 947–963 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. De Moor, B., David, J.: Total linear least squares and the algebraic Riccati equation. Syst. Control Lett. 18, 329–337 (1992)

    Article  MATH  Google Scholar 

  5. Diao, H., Cao, Y.: The perturbation bounds for the solution of weighted Kronecker product linear systems using the W-weighted Drazin inverse. J. Appl. Math. Comput. 20, 1–16 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gohberg, I., Koltracht, I.: Mixed, componentwise, and structured condition numbers. SIAM J. Matrix Anal. Appl. 14, 688–704 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gohberg, I., Kaashoek, M.: An inverse spectral problem for rational matrix functions and minimal divisibility. Integral Equ. Oper. Theory 10, 437–465 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  8. Guo, C.-H.: Nonsymmetric algebraic Riccati equations and Wiener-Hopf factorization for M-matrices. SIAM J. Matrix Anal. Appl. 23, 225–242 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Guo, C.-H.: Efficient methods for solving a nonsymmetric algebraic Riccati equation arising in stochastic fluid models. J. Comput. Appl. Math. 192, 353–373 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Guo, C.-H., Higham, N.J.: Iterative solution of a nonsymmetric algebraic Riccati equation. SIAM J. Matrix Anal. Appl. 29, 396–412 (2007)

    Article  MathSciNet  Google Scholar 

  11. Guo, C.-H., Laub, A.J.: On the iterative solution of a class of nonsymmetric algebraic Riccati equations. SIAM J. Matrix Anal. Appl. 22, 376–391 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Guo, X.-X., Lin, W.-W., Xu, S.-F.: A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation. Numer. Math. 103, 393–412 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  14. Juang, J., Lin, W.-W.: Nonsymmetric algebraic Riccati equations and Hamiltonian-like matrices. SIAM J. Matrix Anal. Appl. 20, 228–243 (1999)

    Article  MathSciNet  Google Scholar 

  15. Kang, W., Xiang, H.: Condition numbers with their condition numbers for the weighted Moore-Penrose inverse and the weighted least squares solution. J. Appl. Math. Comput. 22, 95–112 (2006)

    MATH  MathSciNet  Google Scholar 

  16. Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  17. Laub, A.: Invariant subspace methods for the numerical solution of Riccati equations. In: Bittanti, S., Laub, A., Willems, J. (eds.) The Riccati Equation. Springer, Berlin (1991)

    Google Scholar 

  18. Lu, L.-Z.: Solution form and simple iteration of a nonsymmetric algebraic Riccati equation arising in transport theory. SIAM J. Matrix Anal. Appl. 26, 679–685 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Roberts, J.: Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. Int. J. Control 32, 677–687 (1980)

    Article  MATH  Google Scholar 

  20. Williams, D.: A potential-theoretic note on the quadratic Wiener-Hopf equation for Q-matrices. In: Seminar on Probability XVI. Lecture Notes in Mathematics, vol. 920, pp. 91–94. Springer, Berlin (1982)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yiqin Lin.

Additional information

Yiqin Lin is supported by Scientific Research Startup Foundation of Hunan University of Science and Engineering for Young Teacher. Yimin Wei is supported by the National Natural Science Foundation of China under grant 10471027 and Shanghai Education Committee.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lin, Y., Wei, Y. Normwise, mixed and componentwise condition numbers of nonsymmetric algebraic Riccati equations. J. Appl. Math. Comput. 27, 137–147 (2008). https://doi.org/10.1007/s12190-008-0061-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-008-0061-4

Keywords

Mathematics Subject Classification (2000)

Navigation