Improved power methods for computing eigenvalues of dual quaternion Hermitian matrices

TL;DR

This paper introduces improved power methods and a new eigen-decomposition algorithm for dual quaternion Hermitian matrices, significantly enhancing efficiency and accuracy in eigenvalue computations relevant to multi-agent control.

Contribution

It proposes two enhanced power methods based on dual complex adjoint matrices and Aitken extrapolation, and a novel eigen-decomposition algorithm to address limitations of existing methods.

Findings

The new methods achieve faster convergence than the original power method.

The algorithms demonstrate higher accuracy and speed in numerical experiments.

Application to multi-agent formation control validates practical effectiveness.

Abstract

This paper investigates the eigenvalue computation problem of the dual quaternion Hermitian matrix closely related to multi-agent group control. Recently, power method was proposed by Cui and Qi in Journal of Scientific Computing, 100 (2024) to solve such problem. Recognizing that the convergence rate of power method is slow due to its dependence on the eigenvalue distribution, we propose two improved versions of power method based on dual complex adjoint matrices and Aitken extrapolation, named DCAM-PM and ADCAM-PM. They achieve notable efficiency improvements and demonstrate significantly faster convergence. However, power method may be invalid for dual quaternion Hermitian matrices with eigenvalues having identical standard parts but distinct dual parts. To overcome this disadvantage, utilizing the eigen-decomposition properties of dual complex adjoint matrix, we propose a novel algorithm EDDCAM-EA which surpasses the power method in both accuracy and speed. Application to eigenvalue computations of dual quaternion Hermitian matrices in multi-agent formation control and numerical experiments highlight the remarkable accuracy and speed of our proposed algorithms.