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 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 $|\widehat{C}-AB|\leqslant nu|A||B|$.