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(2N+1).
CITATION STYLE
Schönhage, A. (1982). Asymptotically fast algorithms for the numerical muitiplication and division of polynomials with complex coefficients. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 144 LNCS, pp. 3–15). Springer Verlag. https://doi.org/10.1007/3-540-11607-9_1
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