Skip to main content
SpringerLink
Log in

Numerical methods for nonlinear two-parameter eigenvalue problems

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

We introduce nonlinear two-parameter eigenvalue problems and generalize several numerical methods for nonlinear eigenvalue problems to nonlinear two-parameter eigenvalue problems. As a motivation we consider the computation of critical delays of delay-differential equations with multiple delays.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Akinola, R.O., Freitag, M.A., Spence, A.: The computation of Jordan blocks in parameter-dependent matrices. IMA J. Numer. Anal. 34, 955–976 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrew, A.L., Eric Chu, K.-W., Lancaster, P.: Derivatives of eigenvalues and eigenvectors of matrix functions. SIAM J. Matrix Anal. Appl. 14, 903–926 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atkinson, F.V.: Multiparameter Eigenvalue Problems. Academic Press, New York (1972)

    MATH  Google Scholar 

  4. Betcke, T., Voss, H.: A Jacobi–Davidson-type projection method for nonlinear eigenvalue problems. Future Gener. Comput. Syst. 20, 363–372 (2004)

    Article  Google Scholar 

  5. Bohte, Z.: Calculation of the derivative of the determinant. Obzornik Mat. Fiz. 28, 33–50 (1981)

    MathSciNet  MATH  Google Scholar 

  6. Bohte, Z.: Numerical solution of some twoparameter eigenvalue problems. Anton Kuhelj Memorial Volume, Ljubljana, pp. 17–28 (1982)

  7. Cox, D.A., Little, J.B., O’Shea, D.: Using Algebraic Geometry, 2nd edn. Springer, New York (2005)

    MATH  Google Scholar 

  8. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  9. Hochstenbach, M.E., Košir, T., Plestenjak, B.: A Jacobi–Davidson type method for the nonsingular two-parameter eigenvalue problem. SIAM J. Matrix Anal. Appl. 26, 477–497 (2005)

    Article  MATH  Google Scholar 

  10. Hochstenbach, M.E., Muhič, A., Plestenjak, B.: On linearizations of the quadratic two-parameter eigenvalue problems. Linear Algebr. Appl. 436, 2725–2743 (2012)

    Article  MATH  Google Scholar 

  11. Hochstenbach, M.E., Muhič, A., Plestenjak, B.: Jacobi–Davidson methods for polynomial two-parameter eigenvalue problems . J. Comput. Appl. Math. 288, 251–263 (2015)

  12. Hochstenbach, M.E., Plestenjak, B.: A Jacobi–Davidson type method for a right definite two-parameter eigenvalue problem. SIAM J. Matrix Anal. Appl. 24, 392–410 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hochstenbach, M.E., Plestenjak, B.: Harmonic Rayleigh–Ritz extraction for the multiparameter eigenvalue problem. Electron. Trans. Numer. Anal. 29, 81–96 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Jarlebring, E.: The spectrum of delay-differential equations: numerical methods, stability and perturbation. PhD thesis, TU Braunschweig (2008)

  15. Jarlebring, E., Hochstenbach, M.E.: Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations. Linear Algebr. Appl. 431, 369–380 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ji, X.R., Jiang, H., Lee, B.H.K: A generalized Rayleigh quotient iteration for coupled eigenvalue problems. Technical Report 92-338, Department of Computing and Information Science, Queen’s University (1992)

  17. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)

    Book  MATH  Google Scholar 

  18. Lancaster, P.: Lambda-Matrices and Vibrating Systems. Pergamon Press, Oxford (1966)

    MATH  Google Scholar 

  19. Meerbergen, K., Plestenjak, B.: A Sylvester-Arnoldi type method for the generalized eigenvalue problem with two-by-two operator determinants. Report TW 653, Department of Computer Science, KU Leuven (2014)

  20. Mehrmann, V., Voss, H.: Nonlinear eigenvalue problems: a challenge for modern eigenvalue methods. GAMM Mitt. Ges. Angew. Math. Mech. 27, 121–152 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Muhič, A., Plestenjak, B.: On the singular two-parameter eigenvalue problem. Electron. J. Linear Algebr. 18, 420–437 (2009)

    MathSciNet  MATH  Google Scholar 

  22. Muhič, A., Plestenjak, B.: On the quadratic two-parameter eigenvalue problem and its linearization. Linear Algebr. Appl. 432, 2529–2542 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Neumaier, A.: Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 22, 914–923 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Plestenjak, B.: A continuation method for a right definite two-parameter eigenvalue problem. SIAM J. Matrix Anal. Appl. 21, 1163–1184 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Plestenjak, B.: Numerical methods for the tridiagonal hyperbolic quadratic eigenvalue problem. SIAM J. Matrix Anal. Appl. 28, 1157–1172 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ruhe, A.: Algorithms for the nonlinear eigenvalue problem. SIAM J. Numer. Anal. 10, 674–689 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  27. Schreiber, K.: Nonlinear eigenvalue problems: Newton-type methods and nonlinear Rayleigh functionals. PhD thesis, Department of Mathematics, TU Berlin (2008)

  28. Sleijpen, G.L.G., Booten, A.G.L., Fokkema, D.R., van der Vorst, H.A.: Jacobi–Davidson type methods for generalized eigenproblems and polynomial eigenproblems. BIT 36, 595–633 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Spence, A., Poulton, C.: Photonic band structure calculations using nonlinear eigenvalue techniques. J. Comput. Phys. 204, 65–81 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to the referee for the careful reading, the excellent remarks, and in particular for pointing out the implicit determinant method in [29]. The research was performed in part while the author was visiting the CASA group at the TU Eindhoven. The author wishes to thank the NWO for the visitor grant and the CASA group for the hospitality.

Author information

Authors and Affiliations

  1. IMFM and Department of Mathematics, University of Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia

    Bor Plestenjak

Authors
  1. Bor Plestenjak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bor Plestenjak.

Additional information

Communicated by Daniel Kressner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Plestenjak, B. Numerical methods for nonlinear two-parameter eigenvalue problems. Bit Numer Math 56, 241–262 (2016). https://doi.org/10.1007/s10543-015-0566-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-015-0566-9

Keywords

Mathematics Subject Classification

Search

Navigation