Papers › Fast Evaluation of Real and Complex Polynomials
Fast Evaluation of Real and Complex Polynomials
Ramona Anton, Nicolae Mihalache, François Vigneron
The archive published only this paper's code-link row. Authors, date and abstract are from arXiv's metadata (CC0), read from the Kaggle arXiv metadata snapshot of 2026-09-12 where its title matched the archive's; the title is the archive's.
We propose an algorithm for quickly evaluating polynomials. It pre-conditions a complex polynomial P of degree d in time O(dlogd), 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(√(d(p+logd))) and memory O(dp). The average complexity is computed with respect to points z ∈ℂ, weighted by the spherical area of ℂ. The worst case does not exceed the complexity of H{\"o}rner's scheme. In particular, our algorithm performs asymptotically as O(√(dlogd)) per evaluation. For many classes of polynomials, in particular those with random coefficients in a bounded region of 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.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections