2022/10/30 by Dunster, T. M., Gil, A., Segura, J.
#33C15 #33F05 #34E20 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2210.16982
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 Poincaré asymptotic expansion. We provide numerical evidence showing that the combination of these methods is enough for computing the function with 5× 10-13 relative accuracy in double precision floating point arithmetic.