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Improved power methods for computing eigenvalues of dual quaternion Hermitian matrices

Yongjun Chen, Liping Zhang

TL;DR

The paper addresses the eigenproblem for dual quaternion Hermitian matrices by transforming it to an equivalent problem for dual complex adjoint matrices. It introduces DCAM-PM and ADCAM-PM, which leverage this transformation and Aitken extrapolation to achieve faster convergence than the baseline power method. To overcome limitations when eigenvalues share the same standard part, it develops EDDCAM-EA, an all-eigenpairs algorithm based on dual complex Hermitian eigendecomposition that outperforms the power method in accuracy and speed. Across applications in multi-agent formation control and numerical experiments, the proposed methods demonstrate superior efficiency and robustness, highlighting their potential for broader computational mathematics and engineering use.

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.

Improved power methods for computing eigenvalues of dual quaternion Hermitian matrices

TL;DR

The paper addresses the eigenproblem for dual quaternion Hermitian matrices by transforming it to an equivalent problem for dual complex adjoint matrices. It introduces DCAM-PM and ADCAM-PM, which leverage this transformation and Aitken extrapolation to achieve faster convergence than the baseline power method. To overcome limitations when eigenvalues share the same standard part, it develops EDDCAM-EA, an all-eigenpairs algorithm based on dual complex Hermitian eigendecomposition that outperforms the power method in accuracy and speed. Across applications in multi-agent formation control and numerical experiments, the proposed methods demonstrate superior efficiency and robustness, highlighting their potential for broader computational mathematics and engineering use.

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.

Paper Structure

This paper contains 12 sections, 10 theorems, 83 equations, 1 figure, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Dual complex numbers and dual quaternions
  4. Dual quaternion matrix
  5. Dual Complex Adjoint Matrix
  6. Dual Complex Adjoint Matrix based New Power Method
  7. A Novel Method for Computing All Eigenpairs
  8. Experimental Results
  9. Performance of Aitken Extrapolation
  10. Application in Multi-Agent Formation Control
  11. Addressing Power Method Limitations with the EDDCAM-EA Algorithm
  12. Conclusions

Key Result

lemma thmcounterlemma

Let $\hat{\mathbf P},\hat{\mathbf P}_1\in \hat{\mathbb{Q}}^{m\times k}$, $\hat{\mathbf Q} \in \hat{\mathbb{Q}}^{k\times n}$, $\hat{\mathbf R}\in \hat{\mathbb{Q}}^{n\times n}$, then

Figures (1)

  • Figure 1: Performance of Aitken extrapolation: (a) Convergence of eigenvalue; (b) Convergence of error $e_{\lambda}$.

Theorems & Definitions (19)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • theorem 1
  • proof
  • corollary thmcountercorollary
  • theorem 2
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • ...and 9 more