Algorithm 850: Real parabolic cylinder functions U(a,x), V(a,x)
Fortran 90 programs for the computation of real parabolic cylinder functions are presented. The code computes the functions U( a, x), V( a, x) and their derivatives for real a and x ( x >= 0). The code also computes scaled functions. The range of computation for scaled PCFs is practically unrestricted. The aimed relative accuracy for scaled functions is better than 5 10(-14). Exceptions to this accuracy are the evaluation of the functions near their zeros and the error caused by the evaluation of trigonometric functions of large arguments when |a| >> x. The routines always give values for which the Wronskian relation for scaled functions is verified with a relative accuracy better than 5 10(-14). The accuracy of the unscaled functions is also better than 5 10(-14) for moderate values of x and a ( except close to the zeros), while for large x and a the error is dominated by exponential and trigonometric function evaluations. For IEEE standard double precision arithmetic, the accuracy is better than 5 10(-13) in the computable range of unscaled PCFs ( except close to the zeros).
algorithms parabolic cylinder functions numerical quadrature asymptotic expansions