Prime Numbers

9.5 Large-integer multiplication 487
1. [Initialize]
B =
2b
lg b
;
for(j ∈ [0,D− 1]) {
µj = ωj + 1
;
}
2
θj = ωj − µj; // And so |θj| ≤1/2.
2. [Perform a total of 8B standard FFTs]
for(K ∈ [0, 7]) {
for(β ∈ [0,B− 1]) {
s = (0) D−1
0 ; // Zero signal of length D.
for(j ∈ [0,D− 1]) {
µ = µj mod D;
sµ = sµ + xje−2πiKωj/8θ β
j .
}
FK,β = FFT(s); // So (FK,β,m : m ∈ [0,D− 1]) is a transform.
}
}
3. [Create the transform approximation]
X ′ = ∪ 7 K=0
B−1
β=0
D/8−1 β 1
FK,β,m(−2πim/D) β!
m=0
return X ′ ; // Approximation to the nonuniform DFT (9.23).
Algorithm 9.5.8 is written above in rather compact form, but a typical
implementation on a symbolic processor looks much like this pseudocode.
Note that the signal union at the end—being the left-right concatenation
of length-(D/8) signals—can be effected in some programming languages
about as compactly as we have. Incidentally, though the symbolics mask
somewhat the reason why the algorithm works, it is not hard to see that
the Taylor expansion of e −2πi(µj+θj)k/D in powers of θj, together with adroit
manipulation of indices, brings success. It is a curious and happy fact that
decimating the transform-signal length by the fixed factor of 8 suffices for all
possible input signals x to the algorithm. Such is the utility of the inequality
(9.25). In summary, the number of standard FFTs we require, to yield b-bit
accuracy, is about 16b/ lg b. This is a “worst-case” FFT count, in that practical
applications often enjoy far better than b-bit accuracy when a particular b
parameter is passed to Algorithm 9.5.8.
Since the pioneering work of Dutt–Rokhlin, various works such as [Ware
1998], [Nguyen and Liu 1999] have appeared, revealing somewhat better
accuracy, or slightly improved execution speed, or other enhancements. There
has even been an approach that minimizes the worst-case error for input
signals of unit norm [Fessler and Sutton 2003]. But all the way from the
Dutt–Rokhlin origins to the modern fringe, the basic idea remains the same:
Transform the calculation of X ′ to that of obtaining a set of uniform and
standard FFTs of only somewhat wasteful overall length.