On the evaluation of some sparse polynomials

Nogneng, Dorian; Schost, Éric

doi:10.1090/mcom/3231

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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