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This paper deals with the problem of robot localization from noisy landmark bearings measured by the robot. We present a new localization method which is based on linear constraints, one due to each bearing measurement. This linear system can be solved at low computational cost but yields not very accurate results. Therefore, we transform the system to an equivalent linear system which yields virtually optimal results at a small fraction of the cost of a nonlinear optimization method, which usually achieves the optimal result. Experimental results showing the quality of the results and the low computational cost are presented.