Optimal iterate of the power and inverse iteration methods

The power method is an algorithm for computing the largest eigenvalue of matrix A in absolute value. To find the other eigenvalues one can apply the power method to the matrix −1(A−σI) for some shift σ. This scheme is called the inverse iteration method. Both of these two methods produce a convergence sequence and the limit is approximated by one of the iterates. In the chosen iterate, it may be difficult to estimate the global error, consisting of the truncation error and the round-off error. In this paper, by using the CESTAC method and the CADNA library, we propose a method for computing the optimal iterate, the iterate for which the global error is minimal. In the proposed method the accuracy of the computed eigenvalue may also be estimated. Some numerical examples are given to show the efficiency of the method.