  Open systems with non-Hermitian degeneracies called exceptional points show a
significantly enhanced response to perturbations in terms of large energy
splittings induced by a small perturbation. This reaction can be quantified by
the spectral response strength of the exceptional point. We extend the
underlying theory to the general case where the dimension of the Hilbert space
is larger than the order of the exceptional point. This generalization allows
us to demonstrate an intriguing phenomenon: The spectral response strength of
an exceptional point increases considerably and may even diverge to infinity
under a parameter variation that eventually increases the order of the
exceptional point. This dramatic behavior is in general not accompanied by a
divergence of the energy eigenvalues and is shown to be related to the
well-known divergence of Petermann factors near exceptional points. Finally, an
accurate and robust numerical scheme for the computation of the spectral
response strength based on the general theory and residue calculus is
presented.
