FFT-based Computation of Polynomial Coefficients and Related Tasks

Abstract

We present a FFT-based algorithm for the computation of a polynomial's coefficients from its roots, and apply it to obtain the coefficients of interpolation polynomials, to invert Vandermondians and to evaluate the symmetric functions of a set of parameters. Analytic and numerical evidence with problem sizes up to and beyond n=2000 confirms that it is superior over previous algorithms for these problems in case of parameters taken from uniform or almost uniform distributions on or near circles in the complex plane, and of comparable performance in other cases. Its time complexity is O(n*n) and its storage complexity for tasks other than Vandermondian inversion O(n).