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Accurate complex Jacobi rotations

Vedran Novaković

TL;DR

The paper introduces a high-accuracy method for computing complex Jacobi rotations for the Hermitian 2x2 eigenproblem, deriving a floating-point rotation \tilde{U} from the rounded input using correctly rounded routines \mathtt{cr\_hypot} and \mathtt{cr\_rsqrt}. By scaling the input to avoid subnormal issues and carefully propagating rounding through the rotation parameters, the method achieves tight relative error bounds and outperforms LAPACK's xLAEV2 in key cases. Theoretical guarantees are complemented by extensive numerical testing across datatypes, confirming that the computed rotations remain highly close to unitary and that the observed errors are within derived bounds. Practical impact includes more accurate Jacobi-based EVD computations and improved unitarity in assembled eigenvector matrices for larger Hermitian problems.

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.

Accurate complex Jacobi rotations

TL;DR

The paper introduces a high-accuracy method for computing complex Jacobi rotations for the Hermitian 2x2 eigenproblem, deriving a floating-point rotation \tilde{U} from the rounded input using correctly rounded routines \mathtt{cr\_hypot} and \mathtt{cr\_rsqrt}. By scaling the input to avoid subnormal issues and carefully propagating rounding through the rotation parameters, the method achieves tight relative error bounds and outperforms LAPACK's xLAEV2 in key cases. Theoretical guarantees are complemented by extensive numerical testing across datatypes, confirming that the computed rotations remain highly close to unitary and that the observed errors are within derived bounds. Practical impact includes more accurate Jacobi-based EVD computations and improved unitarity in assembled eigenvector matrices for larger Hermitian problems.

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 and , 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.
Paper Structure (4 sections, 1 theorem, 17 equations, 3 figures)

This paper contains 4 sections, 1 theorem, 17 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Floating-point computation of complex Jacobi rotations
  3. Rounding error analysis
  4. Numerical testing

Key Result

Theorem 1

In at least single precision arithmetic with rounding to nearest, and barring any inexact underflow, the elements of $\widetilde{U}$ computed by xJAEV2 have the relative error bounds as specified, with $\varepsilon$ being the machine precision:

Figures (3)

  • Figure 1: Extremal departures of $\det\widetilde{U}$ from unity in multiples of $\varepsilon$, for ZJAEV2 and ZLAEV2.
  • Figure 2: Observed extremal relative errors, in multiples of $\varepsilon$, in the elements of $\widetilde{U}$ computed by ZJAEV2.
  • Figure 3: Departures from unitarity of the accumulated eigenvector matrices $\mathbf{U}$ of order $n$ from ZJAEVD and ZLAEVD.

Theorems & Definitions (1)

  • Theorem 1