Error bounds on complex floating-point multiplication with an FMAReport as inadecuate

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* Corresponding author 1 LIP - Laboratoire de l-Informatique du Parallélisme 2 ARIC - Arithmetic and Computing Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l-Informatique du Parallélisme 3 IMADA - Department of Mathematics and Computer Science Odense

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 √ 5 u on the normwise relative error | z-z − 1| 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 | z-z − 1| → 2u as u → 0. All our results hold for IEEE floating-point arithmetic, with radix β, precision p, and rounding to nearest; it is only assumed that underflows and overflows do not occur and that $\beta^{p−1} \ge 24$.

Author: Claude-Pierre Jeannerod - Peter Kornerup - Nicolas Louvet - Jean-Michel Muller -



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