  We first consider the problem of approximating a few eigenvalues of a
rational matrix-valued function closest to a prescribed target. It is assumed
that the proper rational part of the rational matrix-valued function is
expressed in the transfer function form $H(s) = C (sI - A)^{-1} B$, where the
middle factor is large, whereas the number of rows of $C$ and the number of
columns of $B$ are equal and small. We propose a subspace framework that
performs two-sided or one-sided projections on the state-space representation
of $H(\cdot)$, commonly employed in model reduction and giving rise to a
reduced transfer function. At every iteration, the projection subspaces are
expanded to attain Hermite interpolation conditions at the eigenvalues of the
reduced transfer function closest to the target, which in turn leads to a new
reduced transfer function. We prove in theory that, when a sequence of
eigenvalues of the reduced transfer functions converges to an eigenvalue of the
full problem, it converges at least at a quadratic rate. In the second part, we
extend the proposed framework to locate the eigenvalues of a general square
large-scale nonlinear meromorphic matrix-valued function $T(\cdot)$, where we
exploit a representation $\mathcal{R}(s) = C(s) A(s)^{-1} B(s) - D(s)$ defined
in terms of the block components of $T(\cdot)$. The numerical experiments
illustrate that the proposed framework is reliable in locating a few
eigenvalues closest to the target point, and that, with respect to runtime, it
is competitive to established methods for nonlinear eigenvalue problems.
