  We describe a novel algorithm for solving general parametric (nonlinear)
eigenvalue problems. Our method has two steps: first, high-accuracy solutions
of non-parametric versions of the problem are gathered at some values of the
parameters; these are then combined to obtain global approximations of the
parametric eigenvalues. To gather the non-parametric data, we use non-intrusive
contour-integration-based methods, which, however, cannot track eigenvalues
that migrate into/out of the contour as the parameter changes. Special
strategies are described for performing the combination-over-parameter step
despite having only partial information on such migrating eigenvalues.
Moreover, we dedicate a special focus to the approximation of eigenvalues that
undergo bifurcations. Finally, we propose an adaptive strategy that allows one
to effectively apply our method even without any a priori information on the
behavior of the sought-after eigenvalues. Numerical tests are performed,
showing that our algorithm can achieve remarkably high approximation accuracy.
