The Influence of Large Nonnormality on the Quality of Convergence of Iterative Methods in Linear Algebra

The departure from normality of a matrix plays an essential role in the numerical matrix computations. The bad numerical behaviour of highly nonnormal matrices has been known for a long time ([14], [25], [5]). But this first effect of high nonnormality i.e. the increase of the spectral instability was considered by practionners as a mathematical oddity, since such matrices were not often encountered in practice. Even the most recent textbooks for engineers on eigenvalue computations, such as [19], do not warn the reader against such a possible difficulty. However, the present-day computers make large-scale problems tractable and allow the engineers to elaborate more and more complex and realistic models for physical phenomena. It seems that now, more and more matrices that model physical problems at the edge of instability arise ([16], [18], [9]), which have a possibly unbounded departure from normality, and they challenge many robust numerical codes because of a second - and newly ana..