Accurate complex Jacobi rotations

TL;DR

This paper presents a method for computing highly accurate complex Jacobi rotations for Hermitian matrices of order two, leveraging correctly rounded functions to ensure high relative accuracy under standard floating-point assumptions.

Contribution

It introduces a novel approach using correctly rounded hypotenuse and reciprocal square root functions to improve the accuracy of complex Jacobi rotations.

Findings

Maximal observed error in rotations' determinants is smaller than LAPACK's.

Method achieves high relative accuracy under mild assumptions.

Numerical examples confirm theoretical error bounds.

Abstract

This note shows how to compute, to high relative accuracy under mild assumptions, complex Jacobi rotations for diagonalization of Hermitian matrices of order two, using the correctly rounded functions $\mathtt{cr\_hypot}$ and $\mathtt{cr\_rsqrt}$, proposed for standardization in the C programming language as recommended by the IEEE-754 floating-point standard. The rounding to nearest (ties to even) and the non-stop arithmetic are assumed. The numerical examples compare the observed with theoretical bounds on the relative errors in the rotations' elements, and show that the maximal observed departure of the rotations' determinants from unity is smaller than that of the transformations computed by LAPACK.