New algorithms for computing the minimum eigenpair of the generalized symmetric eigenvalue problem

Novel methods for computing the minimal eigenvalue of a symmetric positive-definite matrix are presented. The smallest eigenpair (eigenvalue and corresponding eigenvector) of a covariance matrix are computed using the techniques of constrained optimization and higher order root iteration methods. An implementation that relies on QR factorization and less on matrix inversion is presented. This approach can also be used to compute the largest eigenpair by appropriately choosing the initial condition and also can be shown to be applicable to any Hermitian matrix. Several randomly generated test problems are used to evaluate the performance and the computational cost of the methods.