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In some practical applications of array processing, the directions of the incident signals should be estimated adaptively, and/or the time-varying directions should be tracked promptly. In this paper, an adaptive bearing estimation and tracking (ABEST) algorithm is investigated for estimating and tracking the uncorrelated and correlated narrow-band signals impinging on a uniform linear array (ULA) based on the subspace-based method without eigendecomposition (SUMWE), where a linear operator is obtained from the array data to form a basis for the space by exploiting the array geometry and its shift invariance property. Specifically, the space is estimated using the least-mean-square (LMS) or normalized LMS (NLMS) algorithm, and the directions are updated using the approximate Newton method. The transient analyses of the LMS and NLMS algorithms are studied, where the "weight" (i.e., the linear operator) is in the form of a matrix and there is a correlation between the "additive noise" and "input data" that involve the instantaneous correlations of the received array data in the updating equation, and the step-size stability conditions are derived explicitly. In addition, the analytical expressions for the mean-square error (MSE) and mean-square deviation (MSD) learning curves of the LMS algorithm are clarified. The effectiveness of the ABEST algorithm is verified, and the theoretical analyses are corroborated through numerical examples. Simulation results show that the ABEST algorithm is computationally simple and has good adaptation and tracking abilities.