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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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On relative errors of floating-point operations: Optimal bounds and applications
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by Claude-Pierre Jeannerod and Siegfried M. Rump PDF
Math. Comp. 87 (2018), 803-819 Request permission

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 $\mathrm {fl}$ and barring underflow and overflow, such models bound the relative errors $E_1(t) = |t-\mathrm {fl}(t)|/|t|$ and $E_2(t) = |t-\mathrm {fl}(t)|/|\mathrm {fl}(t)|$ 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 $t = x \pm y$ or $t = xy$ with $x,y$ two floating-point numbers. We prove that while this remains true for division in base $\beta > 2$, smaller, attainable bounds can be derived for both division in base $\beta =2$ 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.
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Additional Information
  • Claude-Pierre Jeannerod
  • Affiliation: Inria and Université de Lyon, laboratoire LIP (CNRS, ENS de Lyon, Inria, UCBL), 46 allée d’Italie 69364 Lyon cedex 07, France
  • MR Author ID: 644190
  • Email: claude-pierre.jeannerod@inria.fr
  • Siegfried M. Rump
  • Affiliation: Hamburg University of Technology, Schwarzenbergstraße 95, Hamburg 21071, Germany — and — Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
  • MR Author ID: 151815
  • Email: rump@tuhh.de
  • Received by editor(s): April 20, 2016
  • Received by editor(s) in revised form: October 26, 2016
  • Published electronically: July 7, 2017
  • © Copyright 2017 American Mathematical Society
  • Journal: Math. Comp. 87 (2018), 803-819
  • MSC (2010): Primary 65G50
  • DOI: https://doi.org/10.1090/mcom/3234
  • MathSciNet review: 3739218