Backward error and condition of polynomial eigenvalue problems
The author presents computable expressions for backward errors and condition numbers for the polynomial eigenvalue problem. Different ways to reformulate quadratic eigenvalue problems as generalized eigenvalue problems are considered. It is shown that backward stable algorithms for the generalized eigenvalue problem, e.g.~the QZ algorithm, can be backward unstable for the quadratic eigenvalue problem. For improving the backward error of the solutions obtained by means of the generalized eigenvalue problem formulation the possibility of using a scaling of the quadratic eigenvalue problem is studied. Furthermore, the sensitivity of a given eigenvalue to perturbations in the generalized eigenvalue problem formulations is analyzed. From this analysis one can conclude which formulation is more appropriate for the large and small eigenvalues. Numerical examples confirm this conclusions.
scientific article; zbMATH DE number 1443456
scientific article