Does ill-conditioned mean nearly ill-posed?

                                   by

                       Siegfried M. Rump, Hamburg

                JTB 320 4:30pm, Monday, 20 October, 1997


                                Abstract

Many ill-conditioned problems are nearly ill-posed. This is true for
normwise distances, and it was conjectured in 1991 that, for example for
matrix inversion, this is also true for componentwise relative
perturbations. We prove and extend this conjecture. We show that it is
not satisfied when perturbations are restricted to symmetric
perturbations; also, for least squares problems it is no longer true.
However, an ill-conditioned problem is generally still ill-conditioned
when perturbations are resticted to symmetric perturbations. For the
proof, new concepts in matrix theory are developed, mainly an extension
of Perron-Frobenius theory for matrices without sign restriction.


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