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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机译：如果$ A $是光谱半径小于1的方阵，则$ A ^ k \ to 0 \，{\ text {as}} \，k \ to \ infty $，但是以有限精度算法计算的幂可以是可能不会收敛。我们为$ fl（A ^ k）\ to 0 \，{\ text {as}} \，k \ to \ infty $和$ \ |上的一个边界推导了充分的条件。 fl（A ^ k）\ | $，均以$ A $的约旦典范形式表示。实例表明，结果可能很清晰。我们表明，在某些假设下，当$ A $可对角化时，可以用$ A $的伪谱来重新说明充分条件。我们的分析得出了一条经验法则，可以根据任何后向稳定算法计算出的谱半径小于或大于1来预期$ A $的计算能力的收敛或发散。
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