  Several asymptotic expansions of parabolic cylinder functions are discussed
and error bounds for remainders in the expansions are presented. In particular
Poincar{\'e}-type expansions for large values of the argument $z$ and uniform
expansions for large values of the parameter are considered. The expansions are
based on those given in Olver(1959) and on modifications of these expansions
given in Temme(2000). Computer algebra techniques are used for obtaining
representations of the bounds and for numerical computations.
