Worst-case error analysis for the fast fourier transform

Tight worst-case error bounds are derived for the results of the fast Fourier transform (f.f.t.). The following f.f.t. algorithms are analysed: (i) standard method of complex multiplication without scaling (ii) standard method with prescaling (iii) standard method with stagewise scaling (iv) Buneman´s method of complex multiplication without scaling (v) Buneman´s method with prescaling (vi) Buneman´s method with stagewise scaling. The results establish the maximum number of least-significant bits which can become noisy in the computed spectrum for a given number N of data points, a given accuracy in the arithmetic, and a given accuracy in the tabulated coefficients of the f.f.t. The results are compared with the r.m.s. errors obtained earlier from statistical considerations.2,3 The comparison indicates that the worst-case values cannot be extrapolated from the r.m.s. values, since their dependence on N is different.