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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Error bounds on complex floating-point multiplication


Authors: Richard Brent, Colin Percival and Paul Zimmermann
Journal: Math. Comp. 76 (2007), 1469-1481
MSC (2000): Primary 65G50
Published electronically: January 24, 2007
MathSciNet review: 2299783
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Abstract | References | Similar Articles | Additional Information

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 $ \vert z_0\Vert z_1\vert \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
Email: zimmerma@loria.fr

DOI: http://dx.doi.org/10.1090/S0025-5718-07-01931-X
PII: S 0025-5718(07)01931-X
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)
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.