In terms of a previously suggested Berger et al. (1977) iterative method for finding the lowest-lying eigenstate of hermitian matrixes the problem of pseudo-convergence, i.e. the tendency to converge to an undesired eigenvalue, is discussed. A simple extension of this algorithm is presented which provides a means of identifying and circumventing pseudo-convergence in many cases. This extended scheme may also be used simultaneously to obtain excited states in a numerically economical manner.