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All of the FFT algorithms discussed below compute the DFT exactly ( in exact arithmetic, i. e. neglecting floating-point errors ).
A few " FFT " algorithms have been proposed, however, that compute the DFT approximately, with an error that can be made arbitrarily small at the expense of increased computations.
( 1999 ) achieves lower communication requirements for parallel computing with the help of a fast multipole method.
A wavelet-based approximate FFT by Guo and Burrus ( 1996 ) takes sparse inputs / outputs ( time / frequency localization ) into account more efficiently than is possible with an exact FFT.
Another algorithm for approximate computation of a subset of the DFT outputs is due to Shentov et al.
The Edelman algorithm works equally well for sparse and non-sparse data, since it is based on the compressibility ( rank deficiency ) of the Fourier matrix itself rather than the compressibility ( sparsity ) of the data.
Conversely, if the data are sparse — that is, if only K out of N Fourier coefficients are nonzero — then the complexity can be reduced to O ( K log N log ( N / K )), and this has been demonstrated to lead to practical speedups compared to an ordinary FFT for N / K > 32 in a large-N example ( N = 2 < sup > 22 </ sup >) using a probabilistic approximate algorithm ( which estimates the largest K coefficients to several decimal places ).
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