Testing the Sharpness of Known Error Bounds on the Fast Fourier Transform

The computation of Fast Fourier Transforms (FFTs) in floating-point arithmetic is inexact due to roundings, and for some applications it can prove very useful to know a tight bound on the final error. Although it can be almost attained by specifically built input values, the best known error bound for the Cooley-Tukey FFT seems to be much larger than most actually obtained errors. Also, interval arithmetic can be used to compute a bound on the error committed with a given set of input values, but it is in general considered hampered with large overestimation. We report results of intensive computations to test the two approaches, in order to estimate the numerical performance of state-of-the-art bounds. Surprisingly enough, we observe that while interval arithmetic-based bounds are overestimated, they remain, in our computations, tighter than general known bounds.