Fast algorithm for computating the maximum and minimum Eigenpairs of large matrices

Computing all the eigenvalues and eigenvectors of a large matrix is a time-consuming operation. There are many applications in signal pmssing, control. and applied mathematics that require only the minimum and/or maximum eigenpairs. In this paper, new methods for computing the smallest and largest eigenvalues of symmetric matrix are developed. These methods are modifications of the Rayleigh quotient iteration aimed at citrumventing some drawbacks of that method such as its non or slow convergence. In this approach, the Rayleigh quotient is squentially minimized over several orthogonal vectors. At each iterate, a vector is formed fmm a linear combination of the current iterate and an orthogonal vector that is derived fmm a gradient of a Ritz functional. The pmpsed methods have global and cubic convergence rate. These methods are also generalized to solve high resolution temporal and spatial frequency tracking pmblems. 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. Simulations involving large matrices have shown that the convergence behavior is independent of the size of the matrices.