Sensitivity and Backward Perturbation Analysis of Multiparameter Eigenvalue Problems
Arnab Ghosh, Rafikul Alam · SIAM Journal on Matrix Analysis and Applications · 2018
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 ${cond}(\lambda, \mathbb{W})$ of a simple eigenvalue $\lambda \in \mathbb{C}^m $ of an MEP $\mathbb{W}$ and derive three equivalent representations of ${cond}(\lambda, \mathbb{W})$ of which two are eigenvector-free. Our eigenvector-free representation of ${cond}(\lambda, \mathbb{W})$ provides an alternative viewpoint of the sensitivity of $\lambda.$ We also analyze holomorphic perturbation of a simple eigenvalue of $\mathbb{W}$ when $\mathbb{W}$ varies holomorphically on a parameter $ t \in \mathbb{C}^p.$ For $ \lambda \in \mathbb{C}^m, $ we consider the backward error $\eta(\lambda,\mathbb{W})$ of $\lambda$ as an approximate eigenvalue of $\mathbb{W}$ and determine $\eta(\lambda,\mathbb{W}).$ We construct an optimal perturbation $\Delta \mathbb{W}$ such that $\lambda $ is an eigenvalue of $\mathbb{W}+\Delta \mathbb{W}$ and $|\!|\!| {\Delta \mathbb{W}} |\!|\!| = \eta(\lambda, \mathbb{W}).$ We also consider the backward error $\eta(\lambda,x,\mathbb{W})$ of an approximate eigenpair $(\lambda,x)$ and determine $\eta(\lambda, x,\mathbb{W}).$ Further, we construct an optimal perturbation $\Delta \mathbb{W}$ such that $ \mathbb{W}(\lambda)x + \Delta \mathbb{W}(\lambda)x =0$ and $|\!|\!| {\Delta\mathbb{W}} |\!|\!| =\eta(\lambda, x, \mathbb{W}).$