SENSITIVITY AND BACKWARD PERTURBATION ANALYSIS OF MULTIPARAMETER EIGENVALUE PROBLEMS

We present a general framework for the sensitivity and backward perturbation analysis of linear as well as nonlinear multiparameter eigenvalue problems (MEPs). For a general norm on the space of MEPs, we present a comprehensive analysis of the sensitivity of simple eigenvalues of linear and nonlinear MEPs. We consider the condition number cond(lambda, W) of a simple eigenvalue lambda is an element of C-m of an MEP W and derive three equivalent representations of cond(lambda, W) of which two are eigenvector-free. Our eigenvector-free representation of cond(lambda, W) provides an alternative viewpoint of the sensitivity of lambda. We also analyze holomorphic perturbation of a simple eigenvalue of W when W varies holomorphically on a parameter t is an element of C-P. For lambda is an element of C-m, we consider the backward error eta(lambda, W) of lambda as an approximate eigenvalue of W and determine eta(lambda,W). We construct an optimal perturbation Delta W such that lambda is an eigenvalue of W + Delta W and parallel to Delta W parallel to = eta(lambda,W). We also consider the backward error eta(lambda,x, W) of an approximate eigenpair (lambda, x) and determine eta(lambda, x, W). Further, we construct an optimal perturbation Delta W such that W(lambda)x + Delta W(lambda)x = 0 and parallel to Delta W parallel to = eta(lambda, x, W).