## 2006.166: Matrix powers in finite precision arithmetic

2006.166:
Nicholas J. Higham and Philip A. Knight
(1995)
*Matrix powers in finite precision arithmetic.*
SIAM Journal On Matrix Analysis And Applications, 16 (2).
pp. 343-358.
ISSN 1095-7162

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Official URL: http://locus.siam.org/SIMAX/volume-16/art_0616025.html

## Abstract

If $A$ is a square matrix with spectral radius less than 1 then $A^k \to 0\,{\text{as}}\,k \to \infty $, but the powers computed in finite precision arithmetic may or may not converge. We derive a sufficient condition for $fl( A^k ) \to 0\,{\text{as}}\,k \to \infty $ and a bound on $\| fl ( A^k ) \|$, both expressed in terms of the Jordan canonical form of $A$. Examples show that the results can be sharp. We show that the sufficient condition can be rephrased in terms of a pseudospectrum of $A$ when $A$ is diagonalizable, under certain assumptions. Our analysis leads to the rule of thumb that convergence or divergence of the computed powers of $A$ can be expected according as the spectral radius computed by any backward stable algorithm is less than or greater than 1.

Item Type: | Article |
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Uncontrolled Keywords: | matrix powers, rounding errors, Jordan canonical form, nonnormal matrices, pseudospectrum |

Subjects: | MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis |

MIMS number: | 2006.166 |

Deposited By: | Miss Louise Stait |

Deposited On: | 03 July 2006 |

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