# Improved error bounds for inner products in floating-point arithmetic

* Auteur correspondant
1 ARIC - Arithmetic and Computing
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Given two floating-point vectors $x,y$ of dimension $n$ and assuming rounding to nearest, we show that if no underflow or overflow occurs, any evaluation order for inner product returns a floating-point number $\hat r$ such that $|{\hat r}-x^Ty| \le nu|x|^T|y|$ with $u$ the unit roundoff. This result, which holds for any radix and with no restriction on $n$, can be seen as a generalization of a similar bound given in~\cite{Rump12} for recursive summation in radix $2$, namely $|{\hat r}- x^Te| \le (n-1)u|x|^Te$ with $e=[1,1,\ldots,1]^T$. As a direct consequence, the error bound for the floating-point approximation $\hat C$ of classical matrix multiplication with inner dimension $n$ simplifies to $|\hat{C}-AB|\le nu|A||B|$.
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Article dans une revue
SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2013, 34 (2), pp.338-344. 〈10.1137/120894488〉
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Littérature citée [10 références]

https://hal.inria.fr/hal-00840926
Contributeur : Claude-Pierre Jeannerod <>
Soumis le : mercredi 3 juillet 2013 - 15:10:46
Dernière modification le : vendredi 20 avril 2018 - 15:44:26
Document(s) archivé(s) le : vendredi 4 octobre 2013 - 04:11:05

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JeannerodRump2013.pdf
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Claude-Pierre Jeannerod, Siegfried M. Rump. Improved error bounds for inner products in floating-point arithmetic. SIAM Journal on Matrix Analysis and Applications, Society for Industrial and Applied Mathematics, 2013, 34 (2), pp.338-344. 〈10.1137/120894488〉. 〈hal-00840926〉

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