Prime Numbers

484 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
where g, gH,gW are roots of unity of order WH,H,W, respectively, and the
indices (K + NH)haveK ∈ [0,H− 1], N∈ [0,W − 1]. The last double sum
here can be seen to involve the FFTs of rows and columns of a certain matrix,
as evidenced in the explicit algorithm following:
Algorithm 9.5.7 (Parallel, “four-step” FFT). Let x be a signal of length
D = WH. For algorithmic efficiency we consider the input signal x to be a
matrix T arranged in “columnwise order”; i.e., for j ∈ [0,W − 1] the j-th
column of T contains the (originally contiguous) H elements (xjH+M ) H−1
M=0 . Then,
conveniently, each FFT operation of the overall algorithm occurs on some row of
some matrix (the k-th row vector of a matrix U will be denoted by U (k) ). The
final DFT resides likewise in columnwise order.
1. [H in-place, length-W FFTs, each performed on a row of T ]
for(0 ≤ M