The problem of computing the minimum eigenvector (the Pisarenko vector) of a covariance matrix is of considerable interest in signal processing and computational linear algebra. In this paper, a unified algorithm for computing both the minimum and maximum eigenpairs by simply choosing the proper initial condition is proposed. In particular, the extremum eigenpairs are computed using higher order convergent methods which include the Newton method, the Halley and root iterations. The advantage of these methods is that one can control where the methods converge by choosing a proper initial condition. Some of the methods are implemented using the QR factorization to avoid matrix inversion. By appropriately choosing the initial condition, this approach can also be used to compute the largest eigenpair and thus can be applied for computing the minor and major subspaces of symmetric or hermitian matrices. Procedures such as the double step Newton method for accelerating the developed methods are considered. Several randomly generated test problems are used to evaluate the performance of the methods.