STRUCTURED EIGENVALUE BACKWARD ERRORS OF MATRIX PENCILS AND POLYNOMIALS WITH HERMITIAN AND RELATED STRUCTURES?

摘要

We derive a formula for the backward error of a complex number λ when considered as an approximate eigenvalue of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian, ?-even, and ?-odd. Numerical experiments suggest that in many cases there is a significant difference between the backward errors with respect to perturbations that preserve structure and those with respect to arbitrary perturbations.