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This article presents a new recursive least squares (RLS) adaptive algorithm. The proposed computational scheme uses a finite window by means of a lemma for the system matrix inversion that is, for the first time, stated and proven here. The new algorithm has excellent tracking capabilities. Moreover, its particular structure allows for stabilization by means of a quite simple method. Its stabilized version performs very well not only for a white noise input but also for nonstationary inputs as well. It is shown to follow music, speech, environmental noise, etc., with particularly good tracking properties. The new algorithm can be parallelized via a simple technique. Its parallel form is very fast when implemented with four processors