Computation of maximum and minimum eigenpairs of large matrices

Computing the minimum eigenpair of a matrix is a requirement for many algorithms of signal processing and control. The Rayleigh quotient iteration (RQI) method for locating the minimum eigenpair has a cubic convergence rate and is one of the most efficient algorithms to locate a single eigenpair. However, the RQI method does not always converge to the minimum eigenpair. In this paper efficient techniques for computing the minimum and maximum eigenvalues and their corresponding eigenvectors of Hermitian matrices are developed. The proposed methods exploit the cubic convergence rate of the RQI and ensure convergence to the desired eigenpair for symmetric or hermitian matrices. The core procedure is based on a modified Rayleigh quotient iteration (MRQI) which uses a line search to determine a vector of steepest descent. One of the key features of this method is that it can be implemented in matrix inverse free fashion and thus it is very efficient for very large eigenvalue problems. The proposed algorithms are customized to solve high resolution temporal and spatial frequency tracking problems. The eigenstructure tracking algorithm has update complexity $O(n²p)$, where $n$ is the data dimension and $p$ is the dimension of the minor or major subspaces. The performance of these algorithms is tested with two examples.