  We analyze the magnetic mode structure of axially-magnetized, finite-length,
nanoscopic cylinders in a regime where the exchange interaction dominates,
along with simulations of the mode frequencies of the ferrimagnet yttrium iron
garnet. For the bulk modes we find that the frequencies can be represented by
an expression given by Herring and Kittel by using wavevector components
obtained by fitting the mode patterns emerging from these simulations. In
addition to the axial, radial, and azimuthal modes that are present in an
infinite cylinder, we find localized "cap modes" that are "trapped" at the top
and bottom cylinder faces by the inhomogeneous dipole field emerging from the
ends. Semi-quantitative explanations are given for some of the modes in terms
of a one-dimensional Schrodinger equation which is valid in the exchange
dominant case. The assignment of the azimuthal mode number is carefully
discussed and the frequency splitting of a few pairs of nearly degenerate modes
is determined through the beat pattern emerging from them.
