A Fortran 90 program for the computation of the real parabolic cylinder functions $W(a,\pm x),\,\,x\ge 0$ and their derivatives is presented. The code also computes scaled functions for $a>50$. The functions $W(a,\pm x)$ are a numerically satisfactory pair of solutions of the parabolic cylinder equation $y'' + \left(x^2/4 -a\right)y=0\,,\quad x\ge 0$. Using Wronskian tests, we claim a relative accuracy better than $5\,10^{-13}$ in the computable range of unscaled functions, while for scaled functions the aimed relative accuracy is better than $5\,10^{-14}$. This code, together with the algorithm and related software described in [Gil et al. 2006a],[Gil et al. 2006b] completes the set of software for parabolic cylinder functions (PCFs) for real arguments.

Keywords Parabolic cylinder functions, computation of special functions, ordinary differential equation integration, asymptotic expansions
MSC Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$ (msc 33C15)
Publisher A.C.M.
Journal ACM Transactions on Mathematical Software
Citation
Gil, A, Segura, J, & Temme, N.M. (2011). Algorithm 914: Parabolic Cylinder Function $W(a,x)$ and its Derivative. ACM Transactions on Mathematical Software, 38(1).