Prime Numbers

530 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
This example of exotic transforms, being reminiscent of the discrete Galois
transform (DGT) of the text, appears in [Dimitrov et al. 1995], [Dimitrov et
al. 1998], and has actually been proposed as an idea for obtaining meaningful
spectra—in a discrete field, no less—of real-valued, real-world data.
9.51. Pursuant to Algorithm 9.5.22 for cyclic convolution, work out a similar
algorithm for negacyclic integer convolution via a combined DGT/DWT
method, with halved run length, meaning you want to convolve two real integer
sequences each of length D, via a complex DGT of length D/2. You would need
to establish, for a relevant weighted convolution of length D/2, a (D/2)-th
root of i in a field F p 2 with p a Mersenne prime. Details that may help in such
an implementation can be found in [Crandall 1997b].
9.52. Study the so-called Fermat number transform (FNT) defined by
Xk =
D−1
j=0
xjg −jk (mod fn),
where fn =2 n + 1 and g has multiplicative order D in Zn. A useful choice is
g a power of two, in which case, what are the allowed signal lengths D? The
FNT has the advantage that the internal butterflies of a fast implementation
involve multiply-free arithmetic, but the distinct disadvantage of restricted
signal lengths. A particular question is: Are there useful applications of the
FNT in computational number theory, other than the appearance in the
Schönhage Algorithm 9.5.23?
9.53. In such as Algorithm 9.5.7 one may wish to invoke an efficient
transpose. This is not hard to do if the matrix is square, but otherwise, the
problem is nontrivial. Note that the problem is again trivial, for any matrix,
if one is allowed to copy the original matrix, then write it back in transpose
order. However this can involve long memory jumps, which are not necessary,
as well as all the memory for the copy.
So, work out an algorithm for a general in-place transpose, that is, no
matrix copy allowed, trying to keep everything as “local” as possible, meaning
you want in some sense minimal memory jumps. Some references are [Van
Loan 1992], [Bailey 1990].
9.54. By analyzing the respective complexities of the steps of Algorithm
9.5.8, (1) show that the complexity claim of the text holds for calculating
X ′ k ; (2) give more precise information about the implied big-O constant in
the bound 9.24; and (3) prove the inequality (9.25); and 4) explain how the
inequality leads to the claimed complexity estimate for the algorithm.
The interested reader might investigate/improve on the clever aspect of
the original Dutt–Rokhlin method, which was to expand an oscillation eicz in
a Gaussian series [Dutt and Rokhlin 1993]. There have
9.55. Rewrite Algorithm 9.5.12 to employ balanced-digit representation
(Definition 9.1.2). Note that the important changes center on the carry