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.
ISSN: 0025-5718
Institute: Zuverlässiges Rechnen E-19 
Type: (wissenschaftlicher) Artikel
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