Because the inversion of Abel′s equation often arises in practical contexts, numerous numerical methods for accomplishing it have been proposed. The most successful to-date have been the pseudo-analytic methods of the type proposed by Minerbo & Levy, and Piessens & Verbaeten, as they are simple to implement and conditionally stable. In this paper, we propose methods based on the evaluation of the known inversion formulas using spectral differentiation. We show that they perform as well as the pseudo-analytic methods when the latter behave stably, and that there exist data for which the former yield a satisfactory solution while the latter fail. However, weighing up all numerical considerations, we propose for general implementation an algorithm (viz., Procedure III) which uses a pseudo-analytic method to generate a first approximation to the solution, and a spectral differentiation procedure (viz., Procedure II) to compute the correction to this approximation.