Fast and accurate computation of the Euclidean norm of a vector

Abstract

The numerical computation of the Euclidean norm of a vector is perfectly well conditioned with favorite a priori error estimates. Recently there is interest in computing a faithfully rounded approximation which means that there is no other floating-point number between the computed and the true real result. Hence the result is either the rounded to nearest result or its neighbor. Previous publications guarantee a faithfully rounded result for large dimension, but not the rounded to nearest result. In this note we present several new and fast algorithms producing a faithfully rounded result, as well as the first algorithm to compute the rounded to nearest result. Executable MATLAB codes are included. As a by product, a fast loop-free error-free vector transformation is given. That transforms a vector such that the sum remains unchanged but the condition number of the sum multiplies with the rounding error unit.