  Numerical methods for the computation of the parabolic cylinder $U(a,z)$ for
real $a$ and complex $z$ are presented. The main tools are recent asymptotic
expansions involving exponential and Airy functions, with slowly varying
analytic coefficient functions involving simple coefficients, and stable
integral representations; these two main main methods can be complemented with
Maclaurin series and a Poincar\'e asymptotic expansion. We provide numerical
evidence showing that the combination of these methods is enough for computing
the function with $5\times 10^{-13}$ relative accuracy in double precision
floating point arithmetic.
