Computing the real parabolic cylinder functions U ( a , x ), V ( a , x )

Abstract

C e n t r u m v o o r W i s k u n d e e n I n f o r m a t i c a MAS Modelling, Analysis and Simulation Modelling, Analysis and SimulationComputing the real parabolic cylinder functions U (a,x), V(a,x) A. Gil, J. Segura, N.M. Temme REPORT MAS-E0612 APRIL 2006 Computing the real parabolic cylinder functions U(a,x), V(a,x)ABSTRACT Methods for the computation of real parabolic cylinder functions U(a,x) and V(a,x) and their derivatives are described. We give details on power series, asymptotic series, recursion and quadrature. A combination of these methods can be used for computing parabolic cylinder functions for unrestricted values of the order a and the variable x except for the overflow/underflow limitations. By factoring the dominant exponential factor, scaled functions can be computed without practical overflow/underflow limitations. In an accompanying paper we describe the precise domains for these methods and we present the Fortran 90 codes for the computation of these functions. Methods for the computation of real parabolic cylinder functions U (a, x) and V (a, x) and their derivatives are described. We give details on power series, asymptotic series, recursion and quadrature. A combination of these methods can be used for computing parabolic cylinder functions for unrestricted values of the order a and the variable x except for the overflow/underflow limitations. By factoring the dominant exponential factor, scaled functions can be computed without practical overflow/underflow limitations. In an accompanying paper we describe the precise domains for these methods and we present the Fortran 90 codes for the computation of these functions.

Basic Methods for Computing Special Functions

“…The relation between η and λ in (7.48) becomes clear when we expand 50) and in fact the relation in (7.48) can also be written as 51) where the sign of the square root is positive for λ > 0. For complex values we use analytic continuation.…”

“…Airy-type expansions are used in software for Bessel functions in [4,46,67], and for parabolic cylinder functions in [49,50,91].…”