Error bounds on complex floating-point multiplication with an FMA - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Article Dans Une Revue Mathematics of Computation Année : 2016

Error bounds on complex floating-point multiplication with an FMA

Résumé

The accuracy analysis of complex floating-point multiplication done by Brent, Percival, and Zimmermann [{\it Math.~ Comp.}, 76:1469--1481, 2007] is extended to the case where a fused multiply-add (FMA) operation is available. Considering floating-point arithmetic with rounding to nearest and unit roundoff $u$, we show that their bound $\sqrt 5 \, u$ on the normwise relative error $|\hat z/z-1|$ of a complex product $z$ can be decreased further to $2u$ when using the FMA in the most naive way. Furthermore, we prove that the term $2u$ is asymptotically optimal not only for this naive FMA-based algorithm, but also for two other algorithms, which use the FMA operation as an efficient way of implementing rounding error compensation. Thus, although highly accurate in the componentwise sense, these two compensated algorithms bring no improvement to the normwise accuracy $2u$ already achieved using the FMA naively. Asymptotic optimality is established for each algorithm thanks to the explicit construction of floating-point inputs for which we prove that the normwise relative error then generated satisfies $|\hat z/z-1| \to 2u$ as $u\to 0$. All our results hold for IEEE floating-point arithmetic, with radix $\beta$, precision $p$, and rounding to nearest; it is only assumed that underflows and overflows do not occur and that $\beta^{p-1} \ge 24$.
Fichier principal
Vignette du fichier
JeKoLoMu15.pdf (385.68 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-00867040 , version 1 (27-09-2013)
hal-00867040 , version 2 (12-12-2013)
hal-00867040 , version 3 (25-07-2014)
hal-00867040 , version 4 (16-05-2015)
hal-00867040 , version 5 (04-01-2017)

Identifiants

Citer

Claude-Pierre Jeannerod, Peter Kornerup, Nicolas Louvet, Jean-Michel Muller. Error bounds on complex floating-point multiplication with an FMA. Mathematics of Computation, 2016, ⟨10.1090/mcom/3123⟩. ⟨hal-00867040v4⟩
743 Consultations
697 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More