Fast algorithm for computing 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 processing, control, and applied mathematics that require only the minimum and/or maximum eigenpairs. New methods for computing the smallest and largest eigenvalues of a symmetric matrix are developed. These methods are modifications of the Rayleigh quotient iteration aimed at circumventing some drawbacks of that method such as its non or slow convergence. In this approach, the Rayleigh quotient is sequentially minimized over several orthogonal vectors. At each iteration, a vector is formed from a linear combination of the current iteration and an orthogonal vector that is derived from a gradient of a Ritz functional. The proposed methods have global and cubic convergence rate. These methods are also generalized 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. Simulations involving large matrices have shown that the convergence behavior is independent of the size of the matrices.