A method for locating the minimum eigenvalue and its corresponding eigenvector is considered. The core procedure utilized is the modified Rayleigh quotient iteration (MRQI). The convergence rate of the Rayleigh quotient iteration (RQI) is cubic. However, unfortunately, the RQI may not always locate the minimum eigenvalue. In this paper, a new MRQI that can always locate the minimum eigenpair is given. Based on the MRQI, a fast algorithm to locate minimum eigenpair will be proposed. This method has the following characteristics. First, it does not compute the inclusion interval. Second, it works for any Hermitian matrix as well as Toeplitz matrix. Third, it works on matrices having more than one minimum eigenvalue. Fourth, the numerical error of this method is very small. Lastly, it is attractively simple and fast. The convergence rate of this method is asymptotically cubic. MATLAB simulation results show that tills method may outperform other methods, The term MRQI has been already used. Differences in several MRQI methods are discussed. Mathematical properties of the MRQI are investigated.
This research can be effectively applied to diverse field of the signal processing including communication, because the signal space can be efficiently obtained.