Integral representations for computing real parabolic cylinder functions
Numerische Mathematik
Computing the real parabolic cylinder functions U(a, x), V(a, x)
ACM Transactions on Mathematical Software (TOMS)
Algorithm 914: Parabolic cylinder function W(a, x) and its derivative
ACM Transactions on Mathematical Software (TOMS)
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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).