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In this paper, we discuss rank-one eigenvector updating schemes that are appropriate for tracking time-varying, narrow-band signals in noise. We show that significant reductions in computation are achieved by updating the eigenvalue decomposition (EVD) of a reduced rank version of the data covariance matrix, and that reduced rank updating yields a lower threshold breakdown than full rank updating. We also show that previously published eigenvector updating algorithms , , suffer from a linear build-up of roundoff error which becomes significant when large numbers of recursive updates are performed. We then show that exponential weighting together with pairwise Gram Schmidt partial orthogonalization at each update virtually eliminates the build-up of error making the rank-one update a useful numerical tool for recursive updating. Finally, we compare the frequency estimation performance of reduced rank weighted linear prediction and the LMS algorithm.