Polynomial evaluation via the division algorithm the fast Fourier transform revisited

Abstract

A polynomial p(x) can be evaluated at several points x1,...,xm by first constructing a polynomial d(x) which has x1,...,xm as roots, then dividing p(x) by d(x), and finally evaluating the remainder r(x) at x1,...,xm. This method is useful if the coefficient sequence of d(x) can be chosen to be sparse, thus simplifying the construction of r(x). The case m=1 and d(x) = x−x1 is Horner's rule, while the case d(x) = xm−1 yields the fast Fourier transform algorithm.