Asymptotically fast algorithms for the numerical muitiplication and division of polynomials with complex coefficients

Abstract

Multiplication of univariate n-th degree polynomials over ℂ by straight application of FFT's carried out numerically in ℓ-bit precision will require time O(n log n ψ(ℓ)), where ψ(m) bounds the time for multiplication of m-bit integers, e.g. ψ(m) = cm for pointer machines or ψ(m) = cm·log(m+1)·log log(m+2) for multitape Turing machines. Here a new method is presented, based upon long integer multiplication, by which even faster algorithms can be obtained. Under reasonable assumptions (like ℓ≥log(n+1), and on the coefficient size) polynomial multiplication and discrete Fourier transforms of length n and in ℓ-bit precision are possible in time O(ψ (nℓ)), and division of polynomials in O(ψ(n(ℓ+n))). Included is also a new version of integer multiplication mod(2^N+1).