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.

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