Fast Evaluation of Real and Complex Polynomials

TL;DR

This paper introduces a fast, efficient algorithm for evaluating complex polynomials with complexity that scales favorably with degree and precision, outperforming previous methods especially for specific polynomial classes.

Contribution

The authors present a novel polynomial evaluation algorithm with $O(d ext{log}d)$ pre-conditioning time and $O( ext{sqrt}(d(p+ ext{log}d)))$ average evaluation complexity, outperforming existing schemes.

Findings

Algorithm achieves $O( ext{sqrt}(d ext{log}d))$ evaluation time.

Performs better for polynomials with random coefficients or sparsity.

Complexity and accuracy validated through comprehensive benchmarks.

Abstract

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 H{\"o}rner'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.