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Preprints, Working Papers, ... Year : 2020

Fast in-place algorithms for polynomial operations: division, evaluation, interpolation

Pascal Giorgi
Bruno Grenet

Abstract

We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that such problems can be solved in (nearly) the same asymptotic time as fast polynomial multiplication. However, these reductions, even when applied to an in-place variant of fast polynomial multiplication, yield algorithms which require at least a linear amount of extra space for intermediate results. We demonstrate new in-place algorithms for the aforementioned polynomial computations which require only constant extra space and achieve the same asymptotic running time as their out-of-place counterparts. We also provide a precise complexity analysis so that all constants are made explicit, parameterized by the space usage of the underlying multiplication algorithms.
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Dates and versions

lirmm-02493066 , version 1 (27-02-2020)
lirmm-02493066 , version 2 (12-11-2020)
lirmm-02493066 , version 3 (24-11-2020)

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Pascal Giorgi, Bruno Grenet, Daniel S. Roche. Fast in-place algorithms for polynomial operations: division, evaluation, interpolation. 2020. ⟨lirmm-02493066v1⟩
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