Ultimate complexity for numerical algorithms

Joris van Der Hoeven (1, 2) , Grégoire Lecerf (1, 2)

1 CNRS - Centre National de la Recherche Scientifique
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

Most numerical algorithms are designed for single or double precision oating point arithmetic, and their complexity is measured in terms of the total number of oating point operations. The resolution of problems with high condition numbers (e.g. when approaching a singularity or degeneracy) may require higher working precisions, in which case it is important to take the precision into account when doing complexity analyses. In this paper, we propose a new \ultimate complexity" model, which focuses on analyzing the cost of numerical algorithms for \suciently large" precisions. As an example application we will present an ultimately softly linear time algorithm for modular composition of univariate polynomials.

Domaines: Logiciel mathématique [cs.MS]

Joris van Der Hoeven, Grégoire Lecerf. Ultimate complexity for numerical algorithms. ACM Communications in Computer Algebra, 2020, 54 (1), pp.1-13. ⟨10.1145/3419048.3419049⟩. ⟨hal-03013416⟩