Rayleigh Quotient Iterative Method For The Time-Delay Eigenvalue Problems

In this paper, we research the Rayleigh quotient iterative methods for time-delay eigenvalue problems. Based on the linear approximation of matrix-valued functions, the time-delay eigenvalue problems are transformed into the generalized eigenvalue problems, and the Rayleigh quotient iterative method for solving the time-delay eigenvalue problems is presented, and the convergence analysis of this method is given. In order to accelerate the convergence, the inaccurate Rayleigh quotient iterative method for the time-delay eigenvalue problems is developed. Furthermore,considering the phenomenon of root leakage of the Rayleigh quotient iterative method, we present a region-preserving Rayleigh quotient iterative method for time-delay eigenvalue problems at the condition of the small time-delay and we can compute real eigenvalues in the specified intervals.Numerical results show that the proposed methods are efficient.