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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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Error bounds on complex floating-point multiplication
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by Richard Brent, Colin Percival and Paul Zimmermann PDF
Math. Comp. 76 (2007), 1469-1481 Request permission

Abstract:

Given floating-point arithmetic with $t$-digit base-$\beta$ significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values $z_0$ and $z_1$ can be computed with maximum absolute error $|z_0\|z_1| \frac {1}{2} \beta ^{1 - t} \sqrt {5}$. In particular, this provides relative error bounds of $2^{-24} \sqrt {5}$ and $2^{-53} \sqrt {5}$ for IEEE 754 single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for IEEE 754 single and double precision arithmetic.
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Additional Information
  • Richard Brent
  • Affiliation: Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
  • Email: complex@rpbrent.com
  • Colin Percival
  • Affiliation: IRMACS Centre, Simon Fraser University, Burnaby, BC, Canada
  • Email: cperciva@irmacs.sfu.ca
  • Paul Zimmermann
  • Affiliation: INRIA Lorraine/LORIA, 615 rue du Jardin Botanique, F-54602 Villers-lès-Nancy Cedex, France
  • MR Author ID: 273776
  • Email: zimmerma@loria.fr
  • Received by editor(s): November 21, 2005
  • Received by editor(s) in revised form: February 21, 2006
  • Published electronically: January 24, 2007

  • Dedicated: In memory of Erin Brent (1947–2005)
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 76 (2007), 1469-1481
  • MSC (2000): Primary 65G50
  • DOI: https://doi.org/10.1090/S0025-5718-07-01931-X
  • MathSciNet review: 2299783