Prime Numbers

486 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
standard DFT (9.20) for root g = e −2πi/D ; in the latter case we have the
uniform scenario, ωj = j. The remarkable development—already a decade old
but as we say emerging in importance—is that such nonuniform FFTs can be
calculated to absolute precision ɛ in
O
D ln D + D ln 1
ɛ
operations. In a word: This nonuniform FFT method is “about as efficient”
as a standard, uniform FFT. This efficient algorithm has found application in
such disparate fields as N-body gravitational dynamics (after all, multibody
gravity forces can be evaluated as a kind of convolution) and Riemann zetafunction
computations (see, e.g., Exercise 1.62).
For the following algorithm display, we have departed from the literature
in several ways. First, we force indicial sums like (9.23) to run for j, k ∈
[0,D − 1], for book consistency; indeed, many of the literature references
involve equivalent sums for j or k ∈ [−D/2,D/2−1]. Secondly, we have chosen
an algorithm that is fast and robust (in the sense of guaranteed accuracy even
under radical behavior of the input signal), but not necessarily of minimum
complexity. Robustness is, of course, important for rigorous calculations in
computational number theory. Third, we have chosen this particular algorithm
because it does not rely upon special-function evaluations such as Gaussians
or windowing functions. The penalty for all of this streamlining is that we
are required to perform a certain number of standard, length-D FFTs, this
number of FFTs depending only logarithmically on desired accuracy
In what follows, the “error” ɛ means that the true DFT (9.23) Xk and the
calculated one X ′ k from the algorithm below differ according to
|Xk − X ′ D−1
k|≤ɛ |xj|.
We next present an algorithm that computes Xk in (9.23) to within error
ɛ