Is nonnormality a serious computational difficulty in practice?

The departure from normality of a matrix plays an essential role in numerical matrix computations since it rules the spectral instability. But this rst consequence of high nonnormality was for long considered by practioners as a mathematical oddity, since such matrices were not often encountered in practice. It appears now that more and more matrices, which have a possibly unbounded departure from normality, emerge in the modelling of physical problems at the edge of instability. They challenge many robust numerical codes because of a second and recently exposed consequence of nonnormality : the possible deterioration of the backward stability for algorithms (Chaitin-Chatelin and Frayss e (1996)). In this paper, we address the following ve questions : (i) What is the connection between spectral instability and nonnormality ? (ii) What is an appropriate measure of nonnormality ? (iii) Where do highly nonnormal matrices come from ? (iv) What is the innuence of nonnormality on numerical stability in exact arithmetic ? (v) What is its innuence on the reliability of Numerical Software ? Keywords spectral instability, nonnormality, exact arithmetic, nite precision, reliability of numerical software 2 Is nonnormality a serious computational diiculty in practice ?