Polynomial evaluation via the division algorithm the fast Fourier transform revisited

Abstract

A polynomial p(x) can be evaluated at several points x l ,.......,x m by first constructing a polynomial d(x) which has x l ,...,x m as roots, then dividing p(x) by d(x), and finally evaluating the remainder r(x) at x l ,...,x m . 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=l and d(x) = x-x l is Horner's rule, while the case d(x) = x m -l yields the fast Fourier transform algorithm.