A Fast Compensated Algorithm for Computing Givens Rotations

TL;DR

This paper introduces a simple, fast compensated algorithm for computing highly accurate Givens rotations, improving precision with a straightforward correction scheme that remains effective even with less accurate hypotenuse calculations.

Contribution

The paper presents a novel, simplified compensated method for Givens rotations that enhances accuracy without complex derivations or computations.

Findings

Achieves high accuracy with a simple correction scheme

Effective even with less accurate hypotenuse calculations

Simplifies previous complex algorithms

Abstract

We develop a very simple compensated scheme for computing very accurate Givens rotations. The approach is significantly more straightforward than the one in \cite{borges2021fast}, and the derivation leads to a very satisfying algorithm whereby a naively computed Givens rotation can be used to construct a correction to itself. It is also seen that this scheme continues to provide high accuracy even when built on a hypoteneuse calculation that is of lesser accuracy.