  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 extreme
eigenvalue along with the associated eigenvector of the large dual quaternion
Hermitian matrices. Furthermore, a convergence analysis of the Rayleigh
quotient iteration is derived, demonstrating a local convergence rate of at
least cubic, which is faster than the linear convergence rate of the power
method. 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.
