  We are concerned with the computation of the ${\mathcal L}_\infty$-norm for
an ${\mathcal L}_\infty$-function of the form $H(s) = C(s) D(s)^{-1} B(s)$,
where the middle factor is the inverse of a meromorphic matrix-valued function,
and $C(s),\, B(s)$ are meromorphic functions mapping to short-and-fat and
tall-and-skinny matrices, respectively. For instance, transfer functions of
descriptor systems and delay systems fall into this family. We focus on the
case where the middle factor is large-scale. We propose a subspace projection
method to obtain approximations of the function $H$ where the middle factor is
of much smaller dimension. The ${\mathcal L}_\infty$-norms are computed for the
resulting reduced functions, then the subspaces are refined by means of the
optimal points on the imaginary axis where the ${\mathcal L}_\infty$-norm of
the reduced function is attained. The subspace method is designed so that
certain Hermite interpolation properties hold between the largest singular
values of the original and reduced functions. This leads to a locally
superlinearly convergent algorithm with respect to the subspace dimension,
which we prove and illustrate on various numerical examples.
