FFT-based Computation of Polynomial Coefficients and Related Tasks

TL;DR

This paper introduces an FFT-based algorithm for efficiently computing polynomial coefficients from roots, demonstrating superior performance for large problem sizes and various parameter distributions.

Contribution

The paper presents a novel FFT-based method for polynomial coefficient computation from roots, improving efficiency over previous algorithms for large-scale problems.

Findings

Superior performance for large n (up to 2000)

Effective for parameters on or near circles in the complex plane

Time complexity of O(n^2) with low storage requirements

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).