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On Eigenvector Computation and Eigenvalue Reordering for the Non-Hermitian Quaternion Eigenvalue Problem

Zhigang Jia, Meiyue Shao, Yanjun Shao

TL;DR

This work tackles the dense non-Hermitian quaternion eigenproblem by adding robust eigenvector computation and eigenvalue reordering to the quaternion QR framework, and by integrating aggressive early deflation (AED) to accelerate convergence. It develops quaternion Sylvester solvers and a secure eigenvalue swapping mechanism to enable reordering and AED application on quaternion matrices, paired with a concrete method for recovering eigenvectors from the quaternion Schur form. Numerical experiments on random quaternion matrices demonstrate that the proposed methods yield a faster, backward-stable quaternion eigensolver, with AED substantially reducing the number of QR sweeps and overall runtime. The results lay a foundation for high-performance, structure-preserving quaternion eigensolvers and point toward future HPC-oriented quaternion BLAS and large-scale implementations.

Abstract

In this paper we present several additions to the quaternion QR algorithm, including algorithms for eigenvector computation and eigenvalue reordering. A key outcome of the eigenvalue reordering algorithm is that the aggressive early deflation (AED) technique, which significantly enhances the convergence of the QR algorithm, is successfully applied to the quaternion eigenvalue problem. We conduct numerical experiments to demonstrate the efficiency and effectiveness of the proposed algorithms.

On Eigenvector Computation and Eigenvalue Reordering for the Non-Hermitian Quaternion Eigenvalue Problem

TL;DR

This work tackles the dense non-Hermitian quaternion eigenproblem by adding robust eigenvector computation and eigenvalue reordering to the quaternion QR framework, and by integrating aggressive early deflation (AED) to accelerate convergence. It develops quaternion Sylvester solvers and a secure eigenvalue swapping mechanism to enable reordering and AED application on quaternion matrices, paired with a concrete method for recovering eigenvectors from the quaternion Schur form. Numerical experiments on random quaternion matrices demonstrate that the proposed methods yield a faster, backward-stable quaternion eigensolver, with AED substantially reducing the number of QR sweeps and overall runtime. The results lay a foundation for high-performance, structure-preserving quaternion eigensolvers and point toward future HPC-oriented quaternion BLAS and large-scale implementations.

Abstract

In this paper we present several additions to the quaternion QR algorithm, including algorithms for eigenvector computation and eigenvalue reordering. A key outcome of the eigenvalue reordering algorithm is that the aggressive early deflation (AED) technique, which significantly enhances the convergence of the QR algorithm, is successfully applied to the quaternion eigenvalue problem. We conduct numerical experiments to demonstrate the efficiency and effectiveness of the proposed algorithms.

Paper Structure

This paper contains 13 sections, 2 theorems, 28 equations, 2 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Quaternion right eigenvalue problem
  3. Eigenvector computation from the quaternion Schur form
  4. Motivation
  5. Upper triangular Sylvester equations
  6. Eigenvectors of the quaternion Schur form
  7. Eigenvalue swapping and aggressive early deflation
  8. Eigenvalue swapping algorithm
  9. Quaternion aggressive early deflation
  10. Numerical experiments
  11. Performance tests
  12. Stability tests
  13. Conclusions

Key Result

Lemma 1

Let $\alpha$, $\beta$, $\gamma\in\mathbb H$. Then there exists a unique $\chi\in\mathbb H$ such that $\alpha\chi-\chi\beta=\gamma$ if and only if $\ldbrack\alpha\rdbrack\neq\ldbrack\beta\rdbrack$.

Figures (2)

  • Figure 1: A visual illustration of AED.
  • Figure 2: The number of QR sweeps with respect to the matrix dimension.

Theorems & Definitions (4)

  • Lemma 1: Johnson1944
  • proof
  • Theorem 1
  • proof