The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature

The inversion of the conventional Kontorovich-Lebedev transform with the Macdonald function as kernel and also with a Kelvin function as kernel is considered. Strategies are presented first for computing values of these functions for purely imaginary order (the interior strategy) and these are tested in the subsequent inversion routines (the exterior strategy) which are developed for use where high-speed (mainframe or workstation) computing is available. Both the interior and exterior strategies can exploit the W-transformation and the W-algorithm pioneered by Sidi. It is shown that conventional NAG quadrature software is often incapable of carrying out the inversion without an implementation of this type. Examples considered show that relative error can generally be kept to O(1 · D - 8) although for work requiring hundreds of full inversion computations this would have to be relaxed somewhat to obviate the need for overnight computer runs. Application to a divergent inversion integral, summable in the sense of Abel, is made with satisfactory results. A case with weaker summability is also considered with somewhat inferior results.