Source localization in acoustic waveguides involves a multidimensional search procedure. In this paper, we propose a new algorithm in which the search in the depth direction is replaced by polynomial rooting. Using the proposed algorithm, range and depth estimation by a vertical array requires a 1-D search procedure. For a 3-D localization problem (i.e., range, depth, and direction-of-arrival (DOA) estimation), the algorithm involves a 2-D search procedure. Consequently, the proposed algorithm requires significantly less computation than other methods that are based on a brute-force search procedure over the source location parameters. In order to evaluate the performance of the algorithm, an error analysis is carried out, and Monte-Carlo simulations are performed. The results are compared with the Cramér-Rao bound (CRB) and to the maximum likelihood (ML) simulation performance. The algorithm is shown to be efficient, while being computationally simpler than the ML or the Bartlett processors. The disadvantage of the algorithm is that its SNR threshold occurs in lower SNR than in the ML algorithm.