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Pseudospectra for matrix pencils and stability of equilibria

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Abstract

The concept of ε-pseudospectra for matrices, introduced by Trefethen and his coworkers, has been studied extensively since 1990. In this paper, ε-pseudospectra for matrix pencils, which are relevant in connection with generalized eigenvalue problems, are considered. Some properties as well as the practical computation of ε-pseudospectra for matrix pencils will be discussed. As an application, we demonstrate how this concept can be used for investigating the asymptotic stability of stationary solutions to time-dependent ordinary or partial differential equations; two cases, based on Burgers' equation, will be shown.

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References

  1. L. V. Ahlfors,Complex Analysis.3rd ed., McGraw-Hill, Singapore, 1979.

    MATH  Google Scholar 

  2. T. Braconnier and N. J. Higham,Computing the field of values and pseudospectra using the Lanczos method with continuation, BIT, 36 (1996), pp. 422–440.

    Article  MathSciNet  MATH  Google Scholar 

  3. P. N. Brown and Y. Saad,Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comput., 11 (1990), pp. 450–481.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Brühl,A curve tracing algorithm for computing the pseudospectrum, BIT, 36 (1996), pp. 441–454.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. F. Carpraux, J. Erhel, and M. Sadkane,Spectral portrait for non-Hermitian large sparse matrices, Computing, 53 (1994), pp. 301–310.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. A. Dijkstra, M. J. Molemaker, A. van der Ploeg, and E. F. F. Botta,An efficient code to compute non-parallel steady flows and their linear stability, Computers and Fluids, 24 (1995), pp. 415–434.

    Article  MATH  Google Scholar 

  7. J. L. M. van Dorsselaer, J. F. B. M. Kraaijevanger, and M. N. Spijker,Linear stability analysis in the numerical solution of initial value problems, Acta Numerica (1993), pp. 199–237.

  8. D. R. Fokkema, G. L. G. Sleijpen, and H. A. van der Vorst,Accelerated inexact Newton schemes for large systems of nonlinear equations, Preprint nr. 918, Dept. Math., Utrecht University, Utrecht, 1995. To appear in SIAM J. Sci. Comput.

    Google Scholar 

  9. D. R. Fokkema, G. L. G. Sleijpen, and H. A. van der Vorst,Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils, Preprint nr. 941, Dept. Math., Utrecht University, Utrecht, 1996. To appear in SIAM J. Sci. Comput.

    Google Scholar 

  10. V. Frayssé, M. Gueury, F. Nicoud, and V. Toumazou,Spectral portraits for matrix pencils, Tech. Report TR/PA/96/19, CERFACS, Toulouse, 1996.

    Google Scholar 

  11. A. Greenbaum, V. Pták, and Z. Strakoš,Any nonincreasing convergence curve is possible for GMRES, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 465–469.

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Meerbergen and D. Roose,The restarted Arnoldi method applied to iterative linear system solvers for the computation of rightmost eigenvalues, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. R. Mitchell,Computational Methods in Partial Differential Equations, John Wiley & Sons, London, 1969.

    MATH  Google Scholar 

  14. N. M. Nachtigal, S. C. Reddy, and L. N. Trefethen,How fast are nonsymmetric matrix iterations?, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 778–795.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. C. Reddy and L. N. Trefethen,Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues, Comp. Meth. Appl. Mech. Eng., 80 (1990), pp. 147–164.

    Article  MathSciNet  MATH  Google Scholar 

  16. S. C. Reddy and L. N. Trefethen,Stability of the method of lines, Numer. Math., 62 (1992), pp. 235–267.

    Article  MathSciNet  MATH  Google Scholar 

  17. K. S. Riedel,Generalized epsilon-pseudospectra, SIAM J. Numer. Anal., 31 (1994), pp. 1219–1225.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Ruhe,The rational Krylov algorithm for large nonsymmetric eigenvalues—mapping the resolvent norms (pseudospectrum), Unpublished note (1994).

  19. R. Seydel,Practical Bifurcation and Stability Analysis, 2nd ed., Springer, New York, 1994.

    MATH  Google Scholar 

  20. K.-C. Toh and L. N. Trefethen,Calculation of pseudospectra by the Arnoldi iteration, SIAM J. Sci. Comput., 17 (1996), pp. 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  21. V. Toumazou,Portraits spectraux de matrices: un outil d'analyse de la stabilité, Ph.D. thesis, Université Henri Poincaré, Nancy, 1996.

    Google Scholar 

  22. L. N. Trefethen,Pseudospectra of matrices, in Numerical Analysis 1991, D. F. Griffiths and G. A. Watson, eds., Longman, 1992, pp. 234–266.

  23. L. N. Trefethen,Spectra and Pseudospectra: The Behavior of Non-Normal Matrices and Operators, Book in preparation.

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Authors and Affiliations

  1. Mathematical Institute, Utrecht University, P.O. Box 80.010, NL-3508 TA, Utrecht, The Netherlands

    Jos L. M. Van Dorsselaer

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  1. Jos L. M. Van Dorsselaer
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Communicated by Axel Ruhe

This research has been supported by the Netherlands Organization for Scientific Research (N.W.O.)

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Van Dorsselaer, J.L.M. Pseudospectra for matrix pencils and stability of equilibria. Bit Numer Math 37, 833–845 (1997). https://doi.org/10.1007/BF02510354

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  • DOI: https://doi.org/10.1007/BF02510354

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