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

Authors:
Vu Hoang Linh;Volker Mehrmann;Erik S. Vleck
Affiliations:
Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, Thanh Xuan, Vietnam;Institut für Mathematik, MA 4-5, Technische Universität Berlin, Berlin, Germany 10623;Department of Mathematics, University of Kansas, Lawrence, USA 66045
Venue:
Advances in Computational Mathematics
Year:
2011

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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.