  We present a detailed comparison of numerical solutions of the quasiclassical
Eilenberger equations with several approximation schemes for the density of
states of s- and d-wave superconductors in the vortex state, which have been
used recently. In particular, we critically examine the use of the Doppler
shift method, which has been claimed to give good results for d-wave
superconductors. Studying the single vortex case we show that there are
important contributions coming from core states, which extend far from the
vortex cores into the nodal directions and are not present in the Doppler shift
method, but significantly affect the density of states at low energies. This
leads to sizeable corrections to Volovik's law, which we expect to be sensitive
to impurity scattering. For a vortex lattice we also show comparisons with the
method due to Brandt, Pesch, and Tewordt and an approximate analytical method,
generalizing a method due to Pesch. These are high field approximations
strictly valid close to the upper critical field Bc2. At low energies the
approximate analytical method turns out to give impressively good results over
a broad field range and we recommend the use of this method for studies of the
vortex state at not too low magnetic fields.
