On relative errors of floating-point operations: optimal bounds and applications - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles Mathematics of Computation Year : 2018

On relative errors of floating-point operations: optimal bounds and applications

Abstract

Rounding error analyses of numerical algorithms are most often carried out via repeated applications of the so-called standard models of floating-point arithmetic. Given a round-to-nearest function fl and barring underflow and overflow, such models bound the relative errors E 1 (t) = |t − fl(t)|/|t| and E 2 (t) = |t − fl(t)|/|fl(t)| by the unit roundoff u. This paper investigates the possibility and the usefulness of refining these bounds, both in the case of an arbitrary real t and in the case where t is the exact result of an arithmetic operation on some floating-point numbers. We show that E 1 (t) and E 2 (t) are optimally bounded by u/(1 + u) and u, respectively, when t is real or, under mild assumptions on the base and the precision, when t = x ± y or t = xy with x, y two floating-point numbers. We prove that while this remains true for division in base β > 2, smaller, attainable bounds can be derived for both division in base β = 2 and square root. This set of optimal bounds is then applied to the rounding error analysis of various numerical algorithms: in all cases, we obtain significantly shorter proofs of the best-known error bounds for such algorithms, and/or improvements on these bounds themselves.
Fichier principal
Vignette du fichier
JeannerodRump2016.pdf (391.13 Ko) Télécharger le fichier
Origin Files produced by the author(s)
Loading...

Dates and versions

hal-00934443 , version 1 (22-01-2014)
hal-00934443 , version 2 (21-12-2015)
hal-00934443 , version 3 (20-04-2016)
hal-00934443 , version 4 (03-11-2016)

Identifiers

Cite

Claude-Pierre Jeannerod, Siegfried M. Rump. On relative errors of floating-point operations: optimal bounds and applications. Mathematics of Computation, 2018, 87, pp.803-819. ⟨10.1090/mcom/3234⟩. ⟨hal-00934443v4⟩
1210 View
3354 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More