A New Measure of Relative Error for Vectors

The paper defines a new measure of the relative error of a vector y as an approximation to a vector x, under an arbitrary norm. This is shown to be a metric whereas the traditional measure \|y - x\| / \|x\| is only approximately so. It generalizes F. W. J. Olver’s corresponding measure for scalars, has an explicit formula in inner product spaces, and is proposed as a tool for roundoff error analyses.