Roundoff error analysis of the fast Fourier transform

Abstract

Abstract. This paper presents an analysis of roundoff errors occurring in the floatingpoint computation of the fast Fourier transform. Upper bounds are derived for the ratios of the root-mean-square (RMS) and maximum roundoff errors in the output data to the RMS value of the output data for both single and multidimensional transformations. These bounds are compared experimentally with actual roundoff errors.

Fast Fourier Methods in Computational Complex Analysis

“…Sch6nhage and Strassen (1971) have more sophisticated algorithms than those given here and count operations precisely. Ramos (1971) treats the stability of Fast Fourier Transforms.…”

Generalized FFTs–A survey of some recent results

“…Error analyses of abelian FFTs abound, mainlydue to the widespread use of these algorithms in so many important applications (see e.g. 17,79]). The audience and usefulness of generalized FFTs is growing and analogous analyses for FFTs for arbitrary or speci c classes of groups might be a worthwhile topic to pursue.…”

A stochastic roundoff error analysis for the convolution

“…Since the linear forms in (6) can be decomposed as Xk,p = (Xa)k,p + (Xm)k>p to account separately for the contributions from additions and multiplications, we have The initial conditions for the difference equations (7) and (8) Under the assumption that the a¿)f's are i.i.d. with mean pa and variance <T2, we derive the following bounds on the expected value of the accompanying linear form for addition: (9) .…”

“…Owing to the dimensions of the vectors involved and to the need for repeated computations, the direct evaluation of the convolution product is usually prohibitively expensive [8]. While some studies of the rounding error for the fast Fourier transform can be found in the literature (see [2,6,7,9,12,14]), the issue of the numerical stability of circular convolution is only briefly addressed in [6].…”