Prime Numbers

518 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
There are no more recursive levels (for our example choice b = 2) because
the eval function will break over to some classical method such as an easy
instance of Horner’s rule and evaluate these last a(t),b(t) values directly, each
one at four t = ti values. The final returned entity from eval turns out to be
the sequence
(x(t0),...,x(t7)) = (40320, 518918400, 29654190720, 424097856000,
3100796899200, 15214711438080, 57274321104000, 178462987637760).
Indeed, the product of these eight values is exactly 64!, as expected. One
should note that in such a “product” operation—where evaluations are
eventually all multiplied together—the last phase of the eval function need
not return a union of two signals, but may instead return the product
eval(a, u) ∗ eval(b, v). If that is the designer’s choice, then the step [Check
breakover threshold ...] must also return the product of the indicated x(ti).
Incidentally, polynomial coefficients do not necessarily grow large as the
above example seems to suggest. For one thing, when working on such as a
factoring problem, one will typically be reducing all coefficients modulo some
N, at every level. And there is a clean way to handle the problem of evaluating
x(t) ofdegreeDat some smaller number of points, say at t0,...,tn−1 with
n