  We present a first principles implementation of the dynamic transverse
magnetic susceptibility in the framework of linear response time-dependent
density functional theory. The dynamic susceptibility allows one to obtain the
magnon dispersion as well as magnon lifetimes for a particular material, which
strongly facilitates the interpretation of inelastic neutron scattering
experiments as well as other spectroscopic techniques. We apply the method to
Fe, Ni, and Co and perform a thorough convergence analysis with respect the
basis set size, $k$-point sampling, spectral smearing and unoccupied bands. In
particular, it is shown that while the gap error (acoustic magnon energy at
$\mathbf{q}=\mathbf{0}$) is highly challenging to converge, the spin-wave
stiffness and the dispersion relation itself are much less sensitive to
convergence parameters. Our final results agrees well with experimentally
extracted magnon dispersion relations except for Ni, where it is well-known
that the exchange splitting energy is poorly represented in the local density
approximation. We also find good agreement with previous first principles
calculations and explain how differences in the calculated dispersion relations
can arise from subtle differences in computational approaches.
