Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A structure preserving flow for computing Hamiltonian matrix exponential

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

This article focuses on computing Hamiltonian matrix exponential. Given a Hamiltonian matrix \(\mathcal {H}\), it is well-known that the matrix exponential \(e^{\mathcal {H}}\) is a symplectic matrix and its eigenvalues form reciprocal \((\lambda ,1/\bar{\lambda })\). It is important to take care of the symplectic structure for computing \(e^{\mathcal {H}}\). Based on the structure-preserving flow proposed by Kuo et al. (SIAM J Matrix Anal Appl 37:976–1001, 2016), we develop a numerical method for computing the symplectic matrix pair \((\mathcal {M},\mathcal {L})\) which represents \(e^{\mathcal {H}}\).

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

Access this article

Log in via an institution

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.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Chiang, C.Y., Chu, E.K.-W., Guo, C.H., Huang, T.M., Lin, W.W., Xu, S.F.: Convergence analysis of the doubling algorithm for several nonlinear matrix equations in the critical case. SIAM J. Matrix Anal. Appl. 31, 227–247 (2009)

    Article  MathSciNet  Google Scholar 

  2. Engwerda, J.C., Ran, A.C.M., Rijkeboer, A.L.: Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation \(X + A^* X^{-1}A = Q\). Linear Algebra Appl. 186, 255–275 (1993)

    Article  MathSciNet  Google Scholar 

  3. Hairer, E., Lubich, C., Wanner, G.H.: Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edn. Springer, Berlin (2006)

    MATH  Google Scholar 

  4. Hairer, E., Nørestt, S.P., Wanner, G.: Solving Ordinary Differential Equations I, 2nd edn. Springer, New York (1992)

    Google Scholar 

  5. Higham, N.J.: Functions of matrices: Theory and Computation. SIAM, Philadelphia (2008)

    Book  Google Scholar 

  6. Huang, T.M., Lin, W.W.: Structured doubling algorithms for weakly stabilizing Hermitian solutions of algebraic Riccati equations. Linear Algebra Appl. 430, 1452–1478 (2009)

    Article  MathSciNet  Google Scholar 

  7. Kahan, W., Li, R.-C.: Composition constants for raising the orders of unconventional schemes for ordinary differential equations. Math. Comput. 66, 1089–1099 (1997)

    Article  MathSciNet  Google Scholar 

  8. Kuo, Y.C., Lin, W.W., Shieh, S.F.: Structure-preserving flows of symplectic matrix pairs. SIAM. J. Matrix Anal. Appl. 37, 976–1001 (2016)

    Article  MathSciNet  Google Scholar 

  9. Kuo, Y.C., Lin, W.W., Shieh, S.F.: The asymptotic analysis of the structure-preserving doubling algorithms. Linear Algebra Appl. 531, 318–355 (2017)

    Article  MathSciNet  Google Scholar 

  10. Kuo, Y.C., Shieh, S.F.: A structure-preserving curve for symplectic pairs and its applications. SIAM. J. Matrix Anal. Appl. 33, 597–616 (2012)

    Article  MathSciNet  Google Scholar 

  11. Lancaster, P., Rodman, L.: Algebraic Riccati Equations. Oxford University Press, Oxford (1995)

    MATH  Google Scholar 

  12. Leimkuhler, B.J., Van Vleck, E.S.: Orthosymplectic integration of linear hamiltonian systems. Numer. Math. 77(2), 269–282 (1997)

    Article  MathSciNet  Google Scholar 

  13. Li, R.-C., Kahan, W.: A family of anadromic numerical methods for matrix Riccati differential equations. Math. Comput. 81, 233–265 (2012)

    Article  MathSciNet  Google Scholar 

  14. Lin, W.W., Mehrmann, V., Xu, H.: Canonical forms for Hamiltonian and symplectic matrices and pencils. Linear Algebra Appl. 302–303, 469–533 (1999)

    Article  MathSciNet  Google Scholar 

  15. Lin, W.W., Xu, S.F.: Convergence analysis of structure-preserving doubling algorithms for Riccati-type matrix equations. SIAM J. Matrix Anal. Appl. 28, 26–39 (2006)

    Article  MathSciNet  Google Scholar 

  16. Mehrmann, V.: The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution. Lecture Notes in Control and Inform Sciences, vol. 163. Springer, Berlin (1991)

    Book  Google Scholar 

  17. Mehrmann, V., Poloni, F.: Doubling algorithms with permuted Lagrangian graph bases. SIAM. J. Matrix Anal. Appl. 33, 780–805 (2012)

    Article  MathSciNet  Google Scholar 

  18. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003)

    Article  MathSciNet  Google Scholar 

  19. Schiff, J., Shnider, S.: A natural approach to the numerical integration of riccati differential equations. SIAM J. Numer. Anal. 36(5), 1392–1413 (1999)

    Article  MathSciNet  Google Scholar 

  20. Shieh, S.F., Kuo, Y.C., You, Y.S.: The classification of symplectic matrices and pairs. Ann. Math. Sci. Appl. 4(1), 35–52 (2019)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the Editor, Professor Volker Mehrmann, and the anonymous referees for their careful reading, valuable comments and suggestions on the manuscript.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811, Taiwan

    Yueh-Cheng Kuo

  2. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan

    Wen-Wei Lin

  3. Department of Mathematics, National Taiwan Normal University, Taipei, 116, Taiwan

    Shih-Feng Shieh

Authors
  1. Yueh-Cheng Kuo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wen-Wei Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shih-Feng Shieh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yueh-Cheng Kuo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kuo, YC., Lin, WW. & Shieh, SF. A structure preserving flow for computing Hamiltonian matrix exponential. Numer. Math. 143, 555–582 (2019). https://doi.org/10.1007/s00211-019-01065-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-019-01065-3

Mathematics Subject Classification