We derive a formula for the backward error of a complex number $\lambda$ 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.
Dissipative Hamiltonian (DH) systems are an important concept in energy based modeling of dynamical
systems. One of the major advantages of the DH formulation is that the system encodes system
properties in an algebraic way in the system. Making use of the structure,
it is easy to see that DH systems are stable. In this paper
the question is discussed when a linear constant coefficient DH system is on the boundary
of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues,
or how much it has to be perturbed to be on this boundary. For unstructured systems this distance to instability (stability radius) is well-understood. In this paper,
explicit formulas for this distance under structure preserving perturbations are determined.
It is also shown (via numerical examples) that under structured perturbations the asymptotical
stability of a DH system is much more robust than for unstructured perturbations, since the
distance can be much larger.
We study linear dissipative Hamiltonian (DH) systems with real constant coefficients that arise in energy based modeling of dynamical
systems. In this paper we analyze when such a system is on the boundary
of the region of asymptotic stability, i.e., when it has purely imaginary eigenvalues,
or how much the dissipation term has to be perturbed to be on this boundary. For unstructured systems the explicit construction of the \emph{real distance to instability (real stability radius)} has been a challenging problem. In this paper, we analyze this real distance under different structured perturbations to the dissipation term that preserve the DH structure and we derive explicit formulas for this distance in terms of low rank perturbations. We also show (via numerical examples) that under real structured perturbations to the dissipation the asymptotical
stability of a DH system is much more robust than for unstructured perturbations.