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Recursive least-squares (RLS) algorithms are widely used in many applications, such as real-time signal processing, control and communications. In some applications, regularization of the least-squares provides robustness and enhances performance. Interestingly, updating the regularization parameter as processing data continuously in time is a desirable strategy to improve performance in applications such as beamforming. While many of the presented works in the literature assume non-time-varying regularized RLS (RRLS) techniques, this paper deals with a time-varying RRLS as the parameter varies during the update. The paper proposes a novel and efficient technique that uses an approximate recursive formula, assuming a slight variation in the regularization parameter to provide a low-complexity update method. Simulation results illustrate the feasibility of the derived formula and the superiority of the time-varying RRLS strategy over the fixed one.