New methods for computing the Pisarenko vector

In this paper we show that the Pisarenko vector for harmonic retrieval problems can be obtained without explicit eigendecomposition. The smallest eigenvalue and corresponding eigenvector of a covariance matrix are computed using higher order convergent methods which include the Newton method as special case. 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. Additionally, an approach is proposed to accelerate the developed methods considerably by using the double step Newton method. Several randomly generated test problems are used to evaluate the performance and the computational cost of the methods.