Numerical analysis of Givens rotation

TL;DR

This paper presents a new algorithm for generating 2-by-2 Givens rotation matrices that reduces the number of floating-point operations per entry, leading to increased numerical accuracy despite more total operations.

Contribution

It introduces an algorithm that minimizes operations per entry in Givens rotations, improving accuracy over existing LAPACK methods.

Findings

The new algorithm reduces operations per entry compared to LAPACK.

Numerical tests demonstrate higher average accuracy.

Overall total operations are higher, but per-entry efficiency is improved.

Abstract

Generating 2-by-2 unitary matrices in floating-precision arithmetic is a delicate task. One way to reduce the accumulation error is to use less floating-point operations to compute each of the entries in the 2-by-2 unitary matrix. This paper shows an algorithm that reduces the number of operations to compute the entries of a Givens rotation. Overall, the new algorithm has more operations in total when compared to algorithms in different releases of LAPACK, but less operations per entry. Numerical tests show that the new algorithm is more accurate on average.