A numerical algorithm for the evaluation of Weber parabolic cylinder functions U(a,x), V(a,x), and W(a, ±x)

Weber's parabolic, cylinder functions U(a, x), V(a, x), and W(a, ±x) have recently found wide applications as approximations to quantum mechanical and semi-classical wavefunctions propagating through potential wells or barriers. Available algorithms for their numerical evaluation are inapplicable in some ranges of the two arguments. In this paper we present a, new algorithm, based on the combined use of F. W. J. Olver's (J. Res. Nat. Bur. Stand. Sect. B 63 (1959), 131) uniform asymptotic expansions and E. T. Whittaker's (Proc. London Math. Soc. 35 (1903), 417) complex recurrence relations, to extend their range of usefulness. Using double precision arithmetic, the algorithm generates greater than single precision values of the functions and their derivatives on a Univac 1182.