Practical Divide-and-Conquer Algorithms for Polynomial Arithmetic

  • William Hart
  • Andrew Novocin
Conference paper

DOI: 10.1007/978-3-642-23568-9_16

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6885)
Cite this paper as:
Hart W., Novocin A. (2011) Practical Divide-and-Conquer Algorithms for Polynomial Arithmetic. In: Gerdt V.P., Koepf W., Mayr E.W., Vorozhtsov E.V. (eds) Computer Algebra in Scientific Computing. CASC 2011. Lecture Notes in Computer Science, vol 6885. Springer, Berlin, Heidelberg

Abstract

We investigate two practical divide-and-conquer style algorithms for univariate polynomial arithmetic. First we revisit an algorithm originally described by Brent and Kung for composition of power series, showing that it can be applied practically to composition of polynomials in ℤ[x] given in the standard monomial basis. We offer a complexity analysis, showing that it is asymptotically fast, avoiding coefficient explosion in ℤ[x]. Secondly we provide an improvement to Mulders’ polynomial division algorithm. We show that it is particularly efficient compared with the multimodular algorithm. The algorithms are straightforward to implement and available in the open source FLINT C library. We offer a practical comparison of our implementations with various computer algebra systems.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • William Hart
    • 1
  • Andrew Novocin
    • 2
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.LIP/INRIA/ENSLyon Cedex 07France

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