  We construct approximate solutions that will describe the last stage of
cylindrically symmetric gravitational collapse of dust fluid. Just before the
spacetime singularity formation, the speed of the dust fluid might be almost
equal to the speed of light by gravitational acceleration. Therefore the
analytic solution describing the dynamics of cylindrical null dust might be the
crudest approximate solution of the last stage of the gravitational collapse.
In this paper, we regard this null dust solution as a background and perform
`high-speed approximation' to know the gravitational collapse of ordinary
timelike dust fluid; the `deviation of the timelike
  4-velocity vector field from null' is treated as a perturbation. In contrast
with the null dust approximation, our approximation scheme can describe the
generation of gravitational waves in the course of the cylindrically symmetric
dust collapse.
