A stable high-accuracy method for calculating the inverse of Abel’s integral equation for experimentally derived data is presented. The method employs a piece-wise cubic spline function, least-squares fitted to the data, to represent the function as inverted. Three formulas, two of which are based on the well-known analytic inverses of Abel’s equation and a new one, which was recently developed by the authors and does not contain an explicit derivative, are given for calculating the inverse numerically. The results of numerical tests performed using these formulas on simulated data are presented and compared with the results obtained using other published methods. The comparative study indicates that our method, employing piece-wise least-squares cubic splines, accurately reproduces the inverse function, regardless of the inversion formula employed. It yields markedly better results than do all the other methods compared in this study, when highly error free, sparse, or very noisy data are inverted. The errors at the ends of the data interval (‘‘termination errors’’) are also shown to be smaller than those of the other methods compared in this study.
We present a new inversion formula for Abel’s integral equation which does not require derivatives of any of the functions involved. This is a particularly desirable feature when analyzing experimentally derived data, since differentiation enormously amplifies the random errors inherent in all measured data. The high quality of the results obtainable using the new formula is demonstrated by a typical numerical example for which it yields errors smaller by an order of magnitude than those obtained with the usual inversion formula.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.