Prime Numbers

9.5 Large-integer multiplication 485
whence a final set of row FFTs gives
T =
X0 X2
X1 X3
where Xk =
j xjg −jk are the usual DFT components, and we note that the
final form here for T is again in columnwise order.
Incidentally, if one wonders how this differs from a two-dimensional FFT
such as an FFT in the field of image processing, the answer to that is simple:
This four-step (or six-step, if pre- and post transposes are invoked to start
with and end up with standard row-ordering) format involves that internal
“twist,” or phase factor, in step [Transpose and twist the matrix]. A twodimensional
FFT does not involve the phase-factor twisting step; instead, one
simply takes FFTs of all rows in place, then all columns in place.
Of course, with respect to repeated applications of Algorithm 9.5.7 the
efficient option is simply this: Always store signals and their transforms in
the columnwise format. Furthermore, one can establish a rule that for signal
lengths N =2 n , we factor into matrix dimensions as W = H = √ N =2 n/2 for
n even, but W =2H =2 (n+1)/2 for n odd. Then the matrix is square or almost
square. Furthermore, for the inverse FFT, in which everything proceeds as
above but with FFT −1 calls and the twisting phase uses g +JK , with a final
division by N, one can conveniently assume that the width and height for
this inverse case satisfy W ′ = H ′ or H ′ =2W ′ , so that in such as convolution
problems the output matrix of the forward FFT is what is expected for the
inverse FFT, even when said matrix is nonsquare. Actually, for convolutions
per se there are other interesting optimizations due to J. Papadopoulos, such
as the use of DIF/DIT frameworks and bit-scrambled powers of g; andavery
fast large-FFT implementation of Mayer, in which one never transposes, using
instead a fast, memory-efficient columnwise FFT stage; see [Crandall et al.
1999].
One interesting byproduct of this approach is that one is moved to study
the basic problem of matrix transposition. The treatment in [Bailey 1989] gives
an interesting small example of the algorithm in [Fraser 1976] for efficient
transposition of a stored matrix, while the paper [Van Loan 1992, p. 138]
indicates how active, really, is the ongoing study of fast transpose. Such an
algorithm has applications in other aspects of large-integer arithmetic, for
example see Section 9.5.7.
We next turn to a development that has enjoyed accelerated importance
since its discovery by pioneers A. Dutt and V. Rokhlin. A core result in their
seminal paper [Dutt and Rokhlin 1993] involves a length-D, nonuniform FFT
of the type
Xk =
D−1
j=0
xje −2πikωj/D , (9.23)
where all we know a priori about the (possibly nonuniform) frequencies ωj
is that they all lie in [0,D). This form for Xk is to be compared with the