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The matched-phase coherent broadband matched-field (MF) processor has been previously proposed and shown to outperform other advanced broadband MF processors. It has been previously proposed to search the matched phases using the simulated annealing, which is well known for its ability of solving global optimization problems while having high computational complexity. This prevents simultaneous processing of many frequencies, and thus, limits the processor performance. We propose to use a novel iterative technique, the phase descent search (PDS), for searching the matched phases. This technique is based on coordinate descent optimization which is mainly applicable to solving convex problems. In this work, we investigate its application to the phase search problem, which is a nonconvex problem. We show that the PDS algorithm obtains matched phases similar to that obtained by the simulated annealing, and has significantly lower complexity. Therefore, it enables to search phases for a large number of frequencies and significantly improves the processor performance. The proposed processor is applied to real data from the 1996 Shallow Water Experiment (SWellEx-96) for locating a moving acoustic source at distances between 1 and 9 km with a step of about 150 m. At each distance, one 1-s snapshot with 13 frequencies is enough to provide accurate localization of the source well matched to global positioning system (GPS) measurements.