Fast In-place Algorithms for Polynomial Operations: Division, Evaluation, Interpolation
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.
Cite
@article{arxiv.2002.10304,
title = {Fast In-place Algorithms for Polynomial Operations: Division, Evaluation, Interpolation},
author = {Pascal Giorgi and Bruno Grenet and Daniel S. Roche},
journal= {arXiv preprint arXiv:2002.10304},
year = {2020}
}