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Improved error bounds for inner products in floating-point arithmetic

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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Contributor : Claude-Pierre Jeannerod Connect in order to contact the contributor
Submitted on : Wednesday, July 3, 2013 - 3:10:46 PM
Last modification on : Tuesday, August 2, 2022 - 4:25:10 AM
Long-term archiving on: : Friday, October 4, 2013 - 4:11:05 AM


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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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