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

  • Arnold Schönhage
1. Algorithms I
Part of the Lecture Notes in Computer Science book series (LNCS, volume 144)


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).


Discrete Fourier Transform Algebraic Model Polynomial Multiplication Complex Coefficient Integer Multiplication 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1982

Authors and Affiliations

  • Arnold Schönhage
    • 1
  1. 1.Mathematisches Institut der Universität TübingenTübingenW-Germany

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