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This paper proposes a new fast recursive total least squares (N-RTLS) algorithm to recursively compute the TLS solution for adaptive finite impulse response (FIR) filtering. The N-RTLS algorithm is based on the minimization of the constrained Rayleigh quotient (c-RQ) in which the last entry of the parameter vector is constrained to the negative one. As analysis results on the convergence of the proposed algorithm, we study the properties of the stationary points of the c-RQ. The high computational efficiency of the new algorithm depends on the efficient computation of the fast gain vector (FGV) and the adaptation of the c-RQ. Since the last entry of the parameter vector in the c-RQ has been fixed as the negative one, a minimum point of the c-RQ is searched only along the input data vector, and a more efficient N-RTLS algorithm is obtained by using the FGV. As compared with Davila's RTLS algorithms, the N-RTLS algorithm saves the 6M number of multiplies, divides, and square roots (MADs). The global convergence of the new algorithm is studied by LaSalle's invariance principle. The performances of the relevant algorithms are compared via simulations, and the long-term numerical stability of the N-RTLS algorithm is verified.