Error bounds on complex floating-point multiplication with an FMA (original) (raw)

Résumé

The accuracy analysis of complex floating-point multiplication done by Brent, Percival, and Zimmermann [{\it 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 uuu, we show that their bound sqrt5,u\sqrt 5 \, usqrt5,u on the normwise relative error ∣hatz/z−1∣|\hat z/z-1|∣hatz/z−1∣ of a complex product zzz can be decreased further to 2u2u2u when using the FMA in the most naive way. Furthermore, we prove that the term 2u2u2u 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 2u2u2u 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 ∣hatz/z−1∣to2u|\hat z/z-1| \to 2u∣hatz/z−1∣to2u as uto0u\to 0uto0. All our results hold for IEEE floating-point arithmetic, with radix beta\betabeta, precision ppp, and rounding to nearest; it is only assumed that underflows and overflows do not occur and that betap−1ge24\beta^{p-1} \ge 24betap−1ge24.

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Soumis le : samedi 16 mai 2015-08:57:24

Dernière modification le : mardi 24 février 2026-08:58:02

Archivage à long terme le : jeudi 20 avril 2017-00:23:56

Dates et versions

hal-00867040 , version 1 (27-09-2013)

hal-00867040 , version 2 (12-12-2013)

hal-00867040 , version 3 (25-07-2014)

hal-00867040 , version 4 (16-05-2015)

hal-00867040 , version 5 (04-01-2017)

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Claude-Pierre Jeannerod, Peter Kornerup, Nicolas Louvet, Jean-Michel Muller. Error bounds on complex floating-point multiplication with an FMA. Mathematics of Computation, 2016, ⟨10.1090/mcom/3123⟩. ⟨hal-00867040v4⟩

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