On the Choice of Sign Defining Householder Transformations

TL;DR

This paper discusses how the choice of sign in Householder transformations affects numerical stability, emphasizing that incorrect sign choices can lead to instability depending on data cancellation effects.

Contribution

It highlights the subtle dependence of numerical instability on data cancellation when incorrect signs are used in Householder transformations.

Findings

Incorrect sign choices can cause instability

The extent of instability depends on data cancellation

Proper sign selection is crucial for numerical stability

Abstract

It is well known that, when defining Householder transformations, the correct choice of sign in the standard formula is important to avoid cancellation and hence numerical instability. In this note we point out that when the "wrong" choice of sign is used, the extent of the resulting instability depends in a somewhat subtle way on the data leading to cancellation.