On Euclidean Norm Approximations

TL;DR

This paper analyzes various Euclidean norm approximation methods, comparing their errors and revealing that some previously reported maximum errors are overly optimistic, thereby providing a clearer understanding of their accuracy.

Contribution

It unifies different Euclidean norm approximation methods into a single framework and critically evaluates their error bounds, especially highlighting inaccuracies in prior maximum error estimates.

Findings

Unified mathematical formulation of approximation methods

Comparison of average and maximum errors

Identification of overly optimistic maximum error estimates

Abstract

Euclidean norm calculations arise frequently in scientific and engineering applications. Several approximations for this norm with differing complexity and accuracy have been proposed in the literature. Earlier approaches were based on minimizing the maximum error. Recently, Seol and Cheun proposed an approximation based on minimizing the average error. In this paper, we first examine these approximations in detail, show that they fit into a single mathematical formulation, and compare their average and maximum errors. We then show that the maximum errors given by Seol and Cheun are significantly optimistic.