  We solve few-magnon problems for a finite-size spin-$S$ periodic Heisenberg
XXZ chain with single-ion anisotropy through constructing sets of exact Bloch
states achieving block diagonalization of the system. Concretely, the
two-magnon (three-magnon) problem is converted to a single-particle one on a
one-dimensional (two-dimensional) effective lattice whose size depends linearly
(quadratically) on the total number of sites. For parameters lying within
certain ranges, various types of multimagnon bound states are manifested and
shown to correspond to edge states on the effective lattices. In the absence of
the single-ion anisotropy, we reveal the condition under which exact
zero-energy states emerge. As applications of the formalism, we calculate the
transverse dynamic structure factor for a higher-spin chain near saturation
magnetization and find signatures of the multimagnon bound states. We also
calculate the real-time three-magnon dynamics from certain localized states,
which are relevant to cold-atom quantum simulations, by simulating
single-particle quantum walks on the effective lattices. This provides a
physically transparent interpretation of the observed dynamics in terms of
propagation of bound state excitations. Our method can be directly applied to
more general spin or itinerant particle systems possessing translational
symmetry.
