MATRIX POWERS IN FINITE PRECISION ARITHMETIC

If A is a square matrix with spectral radius less than 1 then A(k) --> 0 as k --> infinity, but the powers computed in finite precision arit hmetic may or may not converge. We derive a sufficient condition for f l(A(k)) --> 0 as k --> infinity and a bound on \\fl(A(k))\\, both expr essed in terms of the Jordan canonical form of A, Examples show that t he 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 tha t convergence or divergence of the computed powers of A can be expecte d according as the spectral radius computed by any backward stable alg orithm is less than or greater than 1.