Minor and major subspace computation of large matrices

Large matrices arise in many formulations in signal processing and control. In this paper, a Rayleigh quotient iteration (RQI) method for locating the minimum eigenpair for symmetric positive definite matrix pencil has been developed. This method has a cubic convergence rate and does not require computation of matrix inversion. The core procedure is based on a modified Rayleigh quotient iteration (MRQI) which uses a line search (exact or approximate) to determine a vector of steepest descent. As a special case, the proposed algorithm is 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 several examples.