Prime Numbers

516 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC
and so require at most three FFTs of that padded length (note that in some
scenarios the y signal’s symmetry allows further optimization).
The third, and most general, case of polynomial evaluation starts from
the observation that polynomial remaindering can be used to decimate the
evaluation procedure. Say that x(t) has degree D − 1 and is to be evaluated
at the points t0,t1,...,tD−1. Let us simplify by assuming that d is a power of
two. If we define two polynomials, each of essentially half the degree of x, by
y0(t) =(t − t0)(t − t1) ...(t − t D/2−1),
y1(t) =(t − t D/2)(t − t D/2+1) ...(t − tD−1),
then we can write the original polynomial in quotient–remainder form as
x(t) =q0(t)y0(t)+r0(t) =q1(t)y1(t)+r1(t).
But this means that a desired evaluation x(tj) iseitherr0(tj) (forj