On Rayleigh Quotient Iteration for Dual Quaternion Hermitian Eigenvalue Problem

TL;DR

This paper develops and analyzes a Rayleigh quotient iteration method for efficiently computing eigenvalues and eigenvectors of dual quaternion Hermitian matrices, which are important in multi-agent formation control.

Contribution

The paper introduces a novel RQI algorithm tailored for dual quaternion Hermitian matrices and provides convergence analysis and numerical validation.

Findings

High accuracy in eigenpair computation

Low CPU time compared to power method

Effective convergence properties

Abstract

The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the Rayleigh quotient iteration (RQI) for solving the right eigenpairs of dual quaternion Hermitian matrices. Combined with dual representation, the RQI algorithm can effectively compute the eigenvalue along with the associated eigenvector of the dual quaternion Hermitian matrices. Furthermore, by utilizing minimal residual property of the Rayleigh Quotient, a convergence analysis of the Rayleigh quotient iteration is derived. Numerical examples are provided to illustrate the high accuracy and low CPU time cost of the proposed Rayleigh quotient iteration compared with the power method for solving the dual quaternion Hermitian eigenvalue problem.