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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bernstein, D.: Multiprecision Multiplication for Mathematicians. In: Accepted by Advances in Applied Mathematics (2001), find at http://cr.yp.to/papers.html#m3
de Boor, C.: B-Form Basics. Geometric Modeling: Algorithms and New Trends, pp. 131–148. SIAM, Philadelphia (1987)
Bostan, A., Salvy, B.: Fast conversion algorithms for orthogonal polynomials (preprint)
Prautzsch, H., Boehm, W., Paluszny, M.: Bézier and B-Spline Techniques. Springer, Heidelberg (2002)
Brent, R., Kung, H.T.: \(\mathcal{O}((n \log n)^3/2)\) Algorithms for composition and reversion of power series. In: Brent, R., Kung, H.T. (eds.) Analytic Computational Complexity, pp. 217–225. Academic Press, New York (1975)
Cannon, J.J., Bosma, W. (eds.): Handbook of Magma Functions, 2.17th edn. (2010), http://magma.maths.usyd.edu.au/magma
von zur Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999)
Hanrot, G., Zimmermann, P.: A long note on Mulder’s short product. Journal of Symbolic Computation 37(3), 391–401 (2004)
Hart, W.: Fast Library for Number Theory: an introduction. In: Fukuda, K., van der Hoeven, J., Joswig, M., Takayama, N. (eds.) ICMS 2010. LNCS, vol. 6327, pp. 88–91. Springer, Heidelberg (2010), http://www.flintlib.org
Knuth, D.: The Art of Computer Programming, volume 2: Seminumerical Algorithms, 3rd edn., pp. 486–488. Addison-Wesley, Reading (1997)
Liu, W., Mann, S.: An analysis of polynomial composition algorithms, University of Waterloo Research Report CS-95-24 (1995)
Mulders, T.: On Short Multiplications and Divisions. In: AAECC, vol. 11, pp. 69–88 (2000)
Pan, V.: Structured matrices and polynomials: unified superfast algorithms, p. 81. Springer, Heidelberg (2001)
Shoup, V.: NTL: A Library for doing Number Theory, open-source library, http://shoup.net/ntl/
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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. https://doi.org/10.1007/978-3-642-23568-9_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-23568-9_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23567-2
Online ISBN: 978-3-642-23568-9
eBook Packages: Computer ScienceComputer Science (R0)