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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On the evaluation of some sparse polynomials
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by Dorian Nogneng and Éric Schost PDF
Math. Comp. 87 (2018), 893-904 Request permission

Abstract:

We give algorithms for the evaluation of sparse polynomials of the form \[ P=p_0 + p_1 x + p_2 x^4 + \cdots + p_{N-1} x^{(N-1)^2},\] for various choices of coefficients $p_i$. First, we take $p_i=p^i$, for some fixed $p$; in this case, we address the question of fast evaluation at a given point in the base ring, and we obtain a cost quasi-linear in $\sqrt {N}$. We present experimental results that show the good behavior of this algorithm in a floating-point context, for the computation of Jacobi theta functions.

Next, we consider the case of arbitrary coefficients; for this problem, we study the question of multiple evaluation: we show that one can evaluate such a polynomial at $N$ values in the base ring in subquadratic time.

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Additional Information
  • Dorian Nogneng
  • Affiliation: LIX, Bâtiment Alan Turing, Campus de l’École Polytechnique, 91120 Palaiseau, France
  • MR Author ID: 1122361
  • Email: dorian.nogneng@lix.polytechnique.fr
  • Éric Schost
  • Affiliation: David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
  • MR Author ID: 672551
  • Email: eschost@uwaterloo.ca
  • Received by editor(s): February 16, 2016
  • Received by editor(s) in revised form: February 23, 2016, and September 16, 2016
  • Published electronically: September 7, 2017
  • © Copyright 2017 American Mathematical Society
  • Journal: Math. Comp. 87 (2018), 893-904
  • MSC (2010): Primary 68W30; Secondary 11Y16
  • DOI: https://doi.org/10.1090/mcom/3231
  • MathSciNet review: 3739222