About Holder condition numbers and the stratification diagram for defective eigenvalues
In this paper, we look at a particular case of application which is Holder continuous, namely the map from a matrix A to one of its eigenvalues lambda, when it is multiple defective. Two asymptotic Holder condition numbers are considered: one (with respect to the other) is associated with a generalization of the Frechet (with respect to Gateaux) derivative [A. Harrabi, About Holder condition numbers of Gateaux and Frechet type for general nonlinear functions, Manuscript - CERFACS, 1998]. We illustrate on a fully developed 3x3 example why these asymptotic condition numbers may not be appropriate for analyzing eigencomputations performed in finite precision. We present the complementary view point of differential geometry, developed by A. Ilahi in [A. Ilahi, Validation du calcul sur ordinateur: application de la theorie des singularites, Ph.D. Thesis, Universite des Sciences Sociales, Toulouse I, 1998], which is based on the stratification diagram in C-3x3. The distance to the stratum indicates when this complementary viewpoint is required. (C) 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.
holder condition number multiple defective eigenvalue index frechet and gateaux derivatives exact arithmetic finite precision arithmetic stratification associated with the commutator ax-xa