Prime Numbers

9.6 Polynomial arithmetic 515
such sums according to
xjT kj = T −k2 /2
xjT −j2 /2
T (−k−j)2 /2
,
j
j
and thus calculates the left-hand sum via the convolution implicit in the righthand
sum. However, in certain settings it is somewhat more convenient to
avoid halving the squares in the exponents, relying instead on properties of
the triangular numbers ∆n = n(n +1)/2. Two relevant algebraic properties
of these numbers are
∆α+β =∆α +∆β + αβ,
∆α =∆−α−1.
A variant of the Bluestein trick can accordingly be derived as
xjT kj = T ∆−k
xjT ∆j T −∆−(k−j) .
j
j
Now the implicit convolution can be performed using only integral powers
of the T constant. Moreover, we can employ an efficient, cyclic convolution
by carefully embedding the x signal in a longer, zero-padded signal and
reindexing, as in the following algorithm.
Algorithm 9.6.6 (Evaluation of polynomial on geometric progression).
Let x(t) = D−1 j=0 xjtj , and let T have an inverse in the arithmetic domain.
This algorithm returns the sequence of values x(T k ) , k ∈ [0,D− 1].
1. [Initialize]
Choose N =2n such that N ≥ 2D;
for(0 ≤ j