A Solution to the Recursive Generalized Eigenvalue Problem

Previously, we have discussed a recursive solution to the vector normal equation utilizing the recursive Cholesky and Modified Gram-Schmidt Orthogonalization (MGSO) algorithms. Previously, we have also discussed a blind adaptive approach for detection and extraction of signals of interest in the presence of noise and interference without relying on preamble or training sequences. The heart of the blind adaptive algorithm is based upon solving the recursive generalized eigenvalue problem, the solution of which is discussed in this paper based on the recursive Cholesky or QR factors and the Householder and QL algorithm with implicit shifts. Even though, the solution to the recursive generalized eigenvalue problem is slightly more efficient than the solution to the direct generalized eigenvalue when all the eigenvalues and eigenvectors are required, the improvement is more noticeable when only one eigenvalue and the corresponding eigenvector are required. Therefore, the solution that is proposed here serves well for recursive generalized eigenvalue problems that require only one eigenvalue and the corresponding eigenvector.