A Quasi-newton Adaptive Algorithm for Estimating Generalized Eigenvectors

We first introduce a constrained minimization formulation for the generalized symmetric eigenvalue problem and then recast it into an unconstrained minimization problem by constructing an appropriate cost function. Minimizer of this cost function corresponds to the eigenvector corresponding to the minimum eigenvalue of the given symmetric matrix pencil and all minimizers are global minimizers. We also present an inflation technique for obtaining multiple generalized eigenvectors of thia pencil. Based on this asymptotic formulation, we derive a quasi-Newton adaptive algorithm for estimating these eigenvectors in the data case. This algorithm is highly modular and parallel with a computational complexity of O(Nz) multiplications, N being the problem-size. Simulation results show fast convergence and good quality of the estimated eigenvectors. 1. I N T R O D U C T I O N Consider the matrix pencil (R,, R,) where R, and R, are N x N symmetric positive definite matrices. Then, the task of computing an N x 1 vector x and a scalar X such that