  A numerical algorithm is proposed to explore in a systematic way the
trajectories of the eigenvalues of non-Hermitian matrices in the parametric
space and exploit this in order to find the locations of defective eigenvalues
in the complex plane. These non-Hermitian degeneracies also called exceptional
points (EP) have raised considerable attention in the scientific community as
these can have a great impact in a variety of physical problems.
  The method requires the computation of successive derivatives of two selected
eigenvalues with respect to the parameter so that, after recombination, regular
functions can be constructed. This algebraic manipulation permits the
localization of exceptional points (EP), using standard root-finding algorithms
and the computation of the associated Puiseux series up to an arbitrary order.
This representation, which is associated with the topological structure of
Riemann surfaces allows to efficiently approximate the selected pair in a
certain neighbourhood of the EP.
  Practical applications dealing with guided acoustic waves propagating in
straight ducts with absorbing walls and in periodic guiding structures are
given to illustrate the versatility of the proposed method and its ability to
handle large size matrices arising from finite element discretization
techniques. The fact that EPs are associated with optimal dissipative
treatments in the sense that they should provide best modal attenuation is also
discussed.
