Publisher DOI: 10.1090/mcom/3234
Title: On relative errors of floating-point operations: Optimal bounds and applications
Language: English
Authors: Jeannerod, Claude Pierre 
Rump, Siegfried M. 
Issue Date: 2018
Source: Mathematics of Computation 310 (87): 803-819 (2018)
Journal or Series Name: Mathematics of computation 
Abstract (english): 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.
URI: http://hdl.handle.net/11420/2654
ISSN: 0025-5718
Institute: Zuverlässiges Rechnen E-19 
Type: (wissenschaftlicher) Artikel
Appears in Collections:Publications without fulltext

Show full item record

Page view(s)

79
Last Week
0
Last month
3
checked on Sep 27, 2020

Google ScholarTM

Check

Add Files to Item

Note about this record

Export

Items in TORE are protected by copyright, with all rights reserved, unless otherwise indicated.