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 r such that | r- xTy| 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, | r- xTe| (n-1)u|x|Te with e=[1,1,…,1]T. As a direct consequence, the error bound for the floating-point approximation C of classical matrix multiplication with inner dimension n simplifies to |C-AB| nu|A||B|.