Fast Evaluation of Real and Complex Polynomials
Résumé
We propose an algorithm for quickly evaluating polynomials. It pre-conditions a complex polynomial $P$ of degree $d$ in time $O(d\log d)$, with a low multiplicative constant independent of the precision. Subsequent evaluations of $P$ computed with a fixed precision of $p$ bits are performed in average arithmetic complexity $O\big(\sqrt{d(p+\log d)}\big)$ and memory $O(dp)$. The average complexity is computed with respect to points $z \in \mathbb{C}$, weighted by the spherical area of $\overline{\mathbb{C}}$. The worst case does not exceed the complexity of Hörner's scheme.
In particular, our algorithm performs asymptotically as $O(\sqrt{d\log d})$ per evaluation. For many classes of polynomials, in particular those with random coefficients in a bounded region of $\mathbb{C}$, or for sparse polynomials, our algorithm performs much better than this upper bound, without any modification or parameterization.
The article contains a detailed analysis of the complexity and a full error analysis, which guarantees that the algorithm performs as well as H\"orner's scheme, only faster. Our algorithm is implemented in a companion library, written in standard C and released as an open-source project [MV22].
Our claims regarding complexity and accuracy are confirmed in practice by a set of comprehensive benchmarks.
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