A Square-Root Free Algorithm for Computing Real Givens Rotations

TL;DR

This paper introduces a new square-root-free algorithm for constructing real Givens rotations, offering improved accuracy and wider applicability on hardware supporting fused multiply-add operations.

Contribution

The paper presents a novel, accurate, square-root-free method for computing Givens rotations that can be directly integrated into various algorithms, unlike previous approaches.

Findings

Algorithm is competitive with hardware square-root methods.

Simulation confirms high accuracy of the proposed method.

Applicable to a wide range of algorithms using Givens rotations.

Abstract

We develop an accurate square-root-free algorithm for constructing real Givens rotations. On processors that support the fused multiply-add operation in hardware, the algorithm is competitive with square-root based algorithms using a hardware square-root. Unlike the square-root-free algorithms in \cite{Hsieh1993,GENTLEMAN1973,Barlow1987,Hammarling1974ANO,Ling1989EfficientLL,Hanson,Hanson2}, our approach will construct the Givens rotation directly and is therefore applicable to a much wider variety of algorithms that use Givens rotations. We investigate the accuracy of the algorithm by simulation.