Improved Error Bounds for Inner Products in Floating-Point Arithmetic

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 an inner product returns a floating-point number ${\widehat r}$ such that $|{\widehat r}- x^Ty| \leqslant nu|x|^T|y|$ with $u$ the unit...

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Veröffentlicht in:SIAM journal on matrix analysis and applications Jg. 34; H. 2; S. 338 - 344
Hauptverfasser: Jeannerod, Claude-Pierre, Rump, Siegfried M.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia Society for Industrial and Applied Mathematics 01.01.2013
Schlagworte:
Approximation
Arithmetic
Computer Arithmetic
Computer Science
Constrictions
Errors
Floating point arithmetic
Mathematical analysis
Numerical Analysis
Product returns
Rounding
Vectors (mathematics)
rounding error analysis
unit in the first place
floating-point inner product
ISSN:0895-4798, 1095-7162
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Zusammenfassung: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 an inner product returns a floating-point number ${\widehat r}$ such that $|{\widehat r}- x^Ty| \leqslant nu|x|^T|y|$ 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 [S. M. Rump, BIT, 52 (2012), pp. 201--220] for recursive summation in radix $2$, namely, $|{\widehat r}- x^Te| \leqslant (n-1)u|x|^Te$ with $e=[1,1,\ldots,1]^T$. As a direct consequence, the error bound for the floating-point approximation $\widehat{C}$ of classical matrix multiplication with inner dimension $n$ simplifies to $|\widehat{C}-AB|\leqslant nu|A||B|$. [PUBLICATION ABSTRACT]
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ISSN:0895-4798
1095-7162
DOI:10.1137/120894488