In this paper a simple iterative algorithm that is guaranteed to produce a stable all-pole filter when minimizing the 1-norm of the linear prediction error signal is proposed. The approach works for both the autocorrelation and covariance frameworks, involves only a one-dimensional search at each step, and obviates the need for linear programming based methods. Based on simulation studies, it was observed that the performance of the algorithm is nearly optimal, i.e., very close to the estimates obtained using interior point methods. Moreover, this method also has the ability to constrain the bandwidth of any peak. The proposed method has been applied for vocal tract estimation and, using spectral distortion as the metric, results are presented using synthetic as well as natural speech. © 2017 Acoustical Society of America.