Accurate Efficient Evaluation of Bessel Transform; Programs and Error Analysis

The method of Filon numerical integration for Fourier transforms is extended to Bessel transforms of a certain form for general gx. Specifically, for the two cases where gx is approximated by straight lines, or parabolas, over abutting panels, the corresponding integrals in the Bessell transform Gomega are evaluated exactly within computer round-off error. Although these integrals cannot be expressed in closed form as for Filons case, a recursive procedure and an asymptotic expansion yield rapid accurate evaluation of the required quantities. Programs are furnished for both cases in BASIC. Furthermore, two versions of each are furnished a faster one requiring considerable storage, and a slower one requiring very little storage. The presence and location of aliasing is predicted and its magnitude is investigated numerically. The error dependence on the panel width used in both cases a and b is established by means of numerical examples, one with a very fast decay with omega, the other with a very slow decay with omega. Keywords Linear approximation Parabolic approximation.