Prime Numbers

488 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
We close this FFT section by mentioning some new developments in regard
to “gigaelement” FFTs that have now become possible at reasonable speed.
For example, [Crandall et al. 2004] discusses theory and implementation for
each of these gigaelement cases:
Length-2 30 , one-dimensional FFT (effected via Algorithm 9.5.7);
2 15 × 2 15 , two-dimensional FFT;
2 10 × 2 10 × 2 10 , three-dimensional FFT.
With such massive signal sizes come the difficult yet fascinating issues of
fast matrix transposition, cache-friendly memory action, and vectorization
of floating-point arithmetic. The bottom line as regards performance is that
the one-dimensional, length-2 30 case takes less than one minute on a modern
hardware cluster, if double-precision floating-point is used. (The two- and
three-dimensional cases are about as fast; in fact the two-dimensional case is
usually fastest, for technical reasons.)
For computational number theory, these new results mean this: On a
hardware cluster that fits into a closet, say, numbers of a billion decimal
digits can be multiplied together in roughly a minute. Such observations
depend on proper resolution of the following problem: The errors in such
“monster” FFTs can be nontrivial. There are just so many terms being
added/multiplied that one deviates from the truth, so to speak, in a kind
of random walk. Interestingly, a length-D FFT can be modeled as a random
walk in D dimensions, having O(ln D) steps. The paper [Crandall et al. 2004]
thus reports quantitative bounds on FFT errors, such bounds having been
pioneered by E. Mayer and C. Percival.
9.5.3 Convolution theory
Let x denote a signal (x0,x1,...,xD−1), where for example, the elements of
x could be the digits of Definitions (9.1.1) or (9.1.2) (although we do not a
priori insist that the elements be digits; the theory to follow is fairly general).
We start by defining fundamental convolution operations on signals. In what
follows, we assume that signals x, y have been assigned the same length (D)
of elements. In all the summations of this next definition, indices i, j each run
over the set {0,...,D− 1}:
Definition 9.5.9. The cyclic convolution of two length-D signals x, y is a
signal denoted z = x × y having D elements given by
zn =
xiyj,
i+j≡n (mod D)
while the negacyclic convolution of x, y is a signal v = x ×− y having D
elements given by
vn =
xiyj −
xiyj,
i+j=n
i+j=D+n