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Multi-point evaluation in higher dimensions

  • Joris van der Hoeven
  • Éric Schost
Article

Abstract

In this paper, we propose efficient new algorithms for multi-dimensional multi-point evaluation and interpolation on certain subsets of so called tensor product grids. These point-sets naturally occur in the design of efficient multiplication algorithms for finite-dimensional \(\mathcal C \)-algebras of the form \(\mathcal A =\mathcal C [x_1, {\ldots }, x_n] / I\), where \(I\) is generated by monomials of the form \(x_1^{i_1} {\cdots } x_n^{i_n}\); one particularly important example is the algebra of truncated power series \(\mathcal C [x_1, {\ldots }, x_n] / (x_1, {\ldots }, x_n)^d\). Similarly to what is known for multi-point evaluation and interpolation in the univariate case, our algorithms have quasi-linear time complexity. As a known consequence Schost (ISSAC’05, ACM, New York, NY, pp 293–300, 2005), we obtain fast multiplication algorithms for algebras \(\mathcal A \) of the above form.

Keywords

Multi-point evaluation Multi-point interpolation Algorithm Complexity Power series multiplication 

Mathematics Subject Classification (2000)

12Y05 68W30 68W40 13P10 65F99 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Laboratoire d’informatiqueUMR 7161 CNRSPalaiseau CedexFrance
  2. 2.Computer Science DepartmentThe University of Western OntarioLondonCanada

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