Backward error and condition of polynomial eigenvalue problems
Françoise Tisseur · Linear Algebra and its Applications · 2000
We develop normwise backward errors and condition numbers for the polynomial eigenvalue problem. The standard way of dealing with this problem is to reformulate it as a generalized eigenvalue problem (GEP). For the special case of the quadratic eigenvalue problem (QEP), we show that solving the QEP by applying the QZ algorithm to a corresponding GEP can be backward unstable. The QEP can be reformulated as a GEP in many ways. We investigate the sensitivity of a given eigenvalue to perturbations in each of the GEP formulations and identify which formulations are to be preferred for large and small eigenvalues, respectively.