Prime Numbers

9.5 Large-integer multiplication 481
while(k ≤ j) {
j = j − k;
k = ⌊k/2⌋;
}
j = j + k;
}
return;
}
It is to be noted that when one performs a convolution in the manner we shall
exhibit later, the scrambling procedures are not needed, provided that one
performs required FFTs in a specific order.
Correct is Gentleman–Sande form (with scrambling procedure omitted)
first, Cooley–Tukey form (without initial scrambling) second. This works out
because, of course, scrambling is an operation of order two.
Happily, in cases where scrambling is not desired, or when contiguous
memory access is important (e.g., on vector computers), there is the Stockham
FFT, which avoids bit-scrambling and also has an innermost loop that runs
essentially consecutively through data memory. The cost of all this is that
one must use an extra copy of the data. The typical implementations of
the Stockham FFT are elegant [Van Loan 1992], but there is a particular
variant that has proved quite useful on modern vector machinery. This special
variant is the “ping-pong” FFT, because one goes back and forth between the
original data and a separate copy. The following algorithm display is based
on a suggested design of [Papadopoulos 1999]:
Algorithm 9.5.6 (FFT, “ping-pong” variant, in-order, no bit-scramble).
Given a (D =2 d )-element signal x, a Stockham FFT is performed, but with
the original x and external data copy y used in alternating fashion. We interpret
X, Y below as pointers to the (complex) signals x, y, respectively, but operating
under the usual rules of pointer arithmetic; e.g., X[0] is the first complex datum
of x initially, but if 4 is added to pointer X, thenX[0] = x4, andsoon.If
exponent d is even, pointer X has the FFT result, else pointer Y has it.
1. [Initialize]
J =1;
X = x; Y = y; // Assign memory pointers.
2. [Outer loop]
for(d ≥ i>0) {
m =0;
while(m 0) {
Y [0] = X[0] + X[D/2];
Y [J] =a(X[0] − X[D/2]);
X = X +1;
Y = Y +1;