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Error bounds on complex floating-point multiplication with an FMA


Authors: Claude-Pierre Jeannerod, Peter Kornerup, Nicolas Louvet and Jean-Michel Muller
Journal: Math. Comp. 86 (2017), 881-898
MSC (2010): Primary 65G50
DOI: https://doi.org/10.1090/mcom/3123
Published electronically: July 15, 2016
MathSciNet review: 3584553
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Abstract: The accuracy analysis of complex floating-point multiplication done by Brent, Percival, and Zimmermann [Math.  Comp., 76:1469-1481, 2007] is extended to the case where a fused multiply-add (FMA) operation is available. Considering floating-point arithmetic with rounding to nearest and unit roundoff $ u$, we show that their bound $ \sqrt 5 \, u$ on the normwise relative error $ \vert\hat z/z-1\vert$ of a complex product $ z$ can be decreased further to $ 2u$ when using the FMA in the most naive way. Furthermore, we prove that the term $ 2u$ is asymptotically optimal not only for this naive FMA-based algorithm but also for two other algorithms, which use the FMA operation as an efficient way of implementing rounding error compensation. Thus, although highly accurate in the componentwise sense, these two compensated algorithms bring no improvement to the normwise accuracy $ 2u$ already achieved using the FMA naively. Asymptotic optimality is established for each algorithm thanks to the explicit construction of floating-point inputs for which we prove that the normwise relative error then generated satisfies $ \vert\hat z/z-1\vert \to 2u$ as $ u\to 0$. All our results hold for IEEE floating-point arithmetic, with radix $ \beta $, precision $ p$, and rounding to nearest; it is only assumed that underflows and overflows do not occur and that $ \beta ^{p-1} \ge 24$.


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Additional Information

Claude-Pierre Jeannerod
Affiliation: Inria, Laboratoire LIP (CNRS, ENS de Lyon, Inria, UCBL), Université de Lyon, 46, allée d’Italie, 69364 Lyon cedex 07, France
Email: claude-pierre.jeannerod@inria.fr

Peter Kornerup
Affiliation: Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
Email: kornerup@imada.sdu.dk

Nicolas Louvet
Affiliation: UCBL, Laboratoire LIP (CNRS, ENS de Lyon, Inria, UCBL), Université de Lyon, 46, allée d’Italie, 69364 Lyon cedex 07, France
Email: nicolas.louvet@ens-lyon.fr

Jean-Michel Muller
Affiliation: CNRS, Laboratoire LIP (CNRS, ENS de Lyon, Inria, UCBL), Université de Lyon, 46, allée d’Italie, 69364 Lyon cedex 07, France
Email: jean-michel.muller@ens-lyon.fr

DOI: https://doi.org/10.1090/mcom/3123
Received by editor(s): September 26, 2013
Received by editor(s) in revised form: July 25, 2014, May 15, 2015, and September 28, 2015
Published electronically: July 15, 2016
Article copyright: © Copyright 2016 American Mathematical Society