A note on homotopic backward error and pseudospectra for the generalized eigenproblem Ax =.pdf

A note on homotopic backward error and pseudospectra forthe generalized eigenproblem Ax = BxV. Toumazou  E. Traviesas yCERFACS Technical Report TR/PA/99/14 - April 1999AbstractWe present the notion of backward error and pseudospectra for homotopic pertur-bations in the case of the generalized eigenproblem Ax = Bx. This is an extensionof the notions introduced in [2] for the standard problem.1 IntroductionIn order to assess the validity of an approximate solution (computed exactly) or acomputed solution (in nite precision), one can use the notion of backward error. Asit was fully described in [2] for the standard problem, in stability analysis, one has todene the data to be perturbed, the type of the perturbations to be considered andthe metrics to measure data and perturbations. The cases of normwise and homotopicperturbations were compared in [2] for the standard problem.For the generalized eigenproblem Ax = Bx, the notion of normwise backward errorhas been studied in [4] and we introduce, in this note, the notions of homotopic back-ward error and pseudospectra.The problem Ax = Bx or equivalently the matrix pair (A;B) is said to be regularif there exists at least one complex number z for which det(A zB) 6= 0. Otherwiseit is said singular. When B is nonsingular, regular matrix pairs (A;B) have the samespectrum as B1A. When B is singular, the matrix pair admits at least one inniteeigenvalue.In this paper, we restrict our study to regular matrix pairs. Therefore, we can ensurethat all the eigenvalues of the matrix pair (A;B) are nite.2 Backward analysis for homotopic perturbationsLet us consider
A and
B homotopic perturbations of A and B
A = tE and
B = tFC.E.R.F.A.C.S, 42, avenue G. Coriolis 31057 Toulouse cedex, toumazou@cerfacs.fryUniversite des Sciences Sociales, Toulouse 1 and C.E.R.F.A.C.S, 42, avenue G. Coriolis 31057 Toulousecedex,travies@cerfacs.fr 1 2 BACKWARD ANALYSIS FOR HOMOTOPIC PERTURBATIONS 2where E and F are given deviation matrices