### Abstract

Language | English |
---|---|

Pages | 234-251 |

Number of pages | 17 |

Journal | ETNA - Electronic Transactions on Numerical Analysis |

Volume | 12 |

Publication status | Published - 2001 |

### Fingerprint

### Keywords

- dynamics
- eigenvalues
- orthogonal iteration
- timestepping
- computer science
- applied mathematics

### Cite this

*ETNA - Electronic Transactions on Numerical Analysis*,

*12*, 234-251.

}

*ETNA - Electronic Transactions on Numerical Analysis*, vol. 12, pp. 234-251.

**Error analysis of QR algorithms for computing Lyapunov exponents.** / McDonald, E.J.; Higham, D.J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Error analysis of QR algorithms for computing Lyapunov exponents

AU - McDonald, E.J.

AU - Higham, D.J.

PY - 2001

Y1 - 2001

N2 - Lyapunov exponents give valuable information about long term dynamics. The discrete and continuous QR algorithms are widely used numerical techniques for computing approximate Lyapunov exponents, although they are not yet supported by a general error analysis. Here, a rigorous convergence theory is developed for both the discrete and continuous QR algorithm applied to a constant coefficient linear system with real distinct eigenvalues. For the discrete QR algorithm, the problem essentially reduces to one of linear algebra for which the timestepping and linear algebra errors uncouple and precise convergence rates are obtained. For the continuous QR algorithm, the stability, rather than the local accuracy, of the timestepping algorithm is relevant, and hence the overall convergence rate is independent of the stepsize. In this case it is vital to use a timestepping method that preserves orthogonality in the ODE system. We give numerical results to illustrate the analysis. Further numerical experiments and a heuristic argument suggest that the convergence properties carry through to the case of complex conjugate eigenvalue pairs.

AB - Lyapunov exponents give valuable information about long term dynamics. The discrete and continuous QR algorithms are widely used numerical techniques for computing approximate Lyapunov exponents, although they are not yet supported by a general error analysis. Here, a rigorous convergence theory is developed for both the discrete and continuous QR algorithm applied to a constant coefficient linear system with real distinct eigenvalues. For the discrete QR algorithm, the problem essentially reduces to one of linear algebra for which the timestepping and linear algebra errors uncouple and precise convergence rates are obtained. For the continuous QR algorithm, the stability, rather than the local accuracy, of the timestepping algorithm is relevant, and hence the overall convergence rate is independent of the stepsize. In this case it is vital to use a timestepping method that preserves orthogonality in the ODE system. We give numerical results to illustrate the analysis. Further numerical experiments and a heuristic argument suggest that the convergence properties carry through to the case of complex conjugate eigenvalue pairs.

KW - dynamics

KW - eigenvalues

KW - orthogonal iteration

KW - timestepping

KW - computer science

KW - applied mathematics

UR - http://www.emis.de/journals/ETNA/vol.12.2001/pp234-251.dir/pp234-251.pdf

M3 - Article

VL - 12

SP - 234

EP - 251

JO - ETNA - Electronic Transactions on Numerical Analysis

T2 - ETNA - Electronic Transactions on Numerical Analysis

JF - ETNA - Electronic Transactions on Numerical Analysis

SN - 1068-9613

ER -