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QR methods and error analysis for computing Lyapunov and Sacker–Sell spectral intervals for linear differential-algebraic equations

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Abstract

In this paper, we propose and investigate numerical methods based on QR factorization for computing all or some Lyapunov or Sacker–Sell spectral intervals for linear differential-algebraic equations. Furthermore, a perturbation and error analysis for these methods is presented. We investigate how errors in the data and in the numerical integration affect the accuracy of the approximate spectral intervals. Although we need to integrate numerically some differential-algebraic systems on usually very long time-intervals, under certain assumptions, it is shown that the error of the computed spectral intervals can be controlled by the local error of numerical integration and the error in solving the algebraic constraint. Some numerical examples are presented to illustrate the theoretical results.

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Authors and Affiliations

  1. Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

    Vu Hoang Linh

  2. Institut für Mathematik, MA 4-5, Technische Universität Berlin, 10623, Berlin, Germany

    Volker Mehrmann

  3. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Erik S. Van Vleck

Authors
  1. Vu Hoang Linh
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  2. Volker Mehrmann
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  3. Erik S. Van Vleck
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Correspondence to Volker Mehrmann.

Additional information

Communicated by Rafael Bru.

This research was supported by Deutsche Forschungsgemeinschaft, through Matheon, the DFG Research Center “Mathematics for Key Technologies” in Berlin.

V.H. Linh’s work was supported by Alexander von Humboldt Foundation and in part by NAFOSTED grant 101.02.63.09; E.S. Van Vleck’s work was supported in part by NSF grants DMS-0513438 and DMS-0812800.

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Linh, V.H., Mehrmann, V. & Van Vleck, E.S. QR methods and error analysis for computing Lyapunov and Sacker–Sell spectral intervals for linear differential-algebraic equations. Adv Comput Math 35, 281–322 (2011). https://doi.org/10.1007/s10444-010-9156-1

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