Convergence in nite precision of successive

This paper is devoted to the analysis of the behaviour, in nite precision arithmetic, of the simplest linear iterative scheme one can think of, that is the successive iteration method (SI) x 0 ; x k+1 = Ax k + b; k 0 where A is a real or complex matrix of order n and x is a real or complex vector of size n. In exact arithmetic, the behaviour of (1) is completely understood ; there is convergence for any x 0 if and only if (A) < 1 where (A) is the spectral radius of A. When (SI) is run on a computer with nite precision arithmetic, then for certain matrices A, the convergence is not guaranteed in practice when (A) < 1 is true in exact arithmetic. It is clear that the phenomenon should be attributed to the conjunction of two factors : i) the nonnormality of A and ii) the nite precision of the computer arithmetic. We perform a straightforward analysis of the convergence of (SI) in nite precision from which we try to understand the subtle interplay between factors i) and ii) which takes place inside the computer, when the iteration matrix A has a high nonnormality. Why should nonnormality be an issue in nite precision ? Because only nonnormal matrices can display a signiicant amount of spectral instability. Therefore a small perturbation A on A can result in a large perturbation of the spectrum. When the spectral instability of A is high, it appears that a convergence condition such as (A) < 1 may not be generic enough for nite precision computations.