Prime Numbers

9.6 Polynomial arithmetic 517
Note that in the calculations of w(t),z(t) the intent is that the product must
be expanded, to render w, z as signals of coefficients. The operations to expand
these products must be taken into account in any proper complexity estimate
for this evaluation algorithm (see Exercise 9.75). Along such lines, note that
an especially efficient way to implement Algorithm 9.6.7 is to preconstruct a
polynomial remainder tree; that is, to exploit the fact that the polynomials
in Step [Assemble half-polynomials] have been calculated from their own
respective halves, and so on.
To lend support to the reader who desires to try this general evaluation
Algorithm 9.6.7, let us give an example of its workings. Consider the task
of calculating the number 64! not by the usual, sequential multiplication of
successive integers but by evaluating the polynomial
x(t) =t(1 + t)(2 + t)(3 + t)(4 + t)(5 + t)(6 + t)(7 + t)
at the 8 points
= 5040t + 13068t 2 + 13132t 3 + 6769t 4 + 1960t 5 322t 6 +28t 7 + t 8
T =(1, 9, 17, 25, 33, 41, 49, 57)
and then taking the product of the eight evaluations to get the factorial.
Since the algorithm is fully recursive, tracing is nontrivial. However, if we
assign b = 2, say, in Step [Set breakover] and print out the half-polynomials
w, z and polynomial-mod results a, b right after these entities are established,
then our output should look as follows. On the first pass of eval we obtain
w(t) = 3825 − 4628t + 854t 2 − 52t 3 ,
z(t) = 3778929 − 350100t + 11990t 2 − 180t 3 + t 4 ,
a(t) =x(t) modw(t)
= −14821569000 + 17447650500t − 2735641440t 2 + 109600260t 3 ,
b(t) =x(t) modz(t)
= −791762564494440 + 63916714435140t − 1735304951520t 2
+ 16010208900t 3 ,
and for each of a, b there will be further recursive passes of eval. If we keep
tracing in this way, the subsequent passes reveal
w(t) =9− 10t + t 2 ,
z(t) = 425 − 42t + t 2 ,
and, continuing in recursive order,
a(t) =−64819440 + 64859760t,
b(t) =−808538598000 + 49305458160t,
w(t) = 1353 − 74t + t 2 ,
z(t) = 2793 − 106t + t 2 ,
a(t) =−46869100573680 + 1514239317360t,
b(t) =−685006261415280 + 15148583316720t.