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arXiv · 2002.10304

Fast In-place Algorithms for Polynomial Operations: Division, Evaluation, Interpolation

Pascal Giorgi·Bruno Grenet·Daniel S. Roche
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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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Pascal Giorgi, Bruno Grenet, Daniel S. Roche. 2020-02-24. Fast In-place Algorithms for Polynomial Operations: Division, Evaluation, Interpolation. https://doi.org/10.1145/3373207.3404061

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