When signal subspace techniques, such as MuSIC, are used to locate a number of incident signals, an exhaustive search of the array manifold has to be carried out. This search involves the evaluation of a single cost function at a number of points which form a grid, resulting in quantization-error effects. In this paper a new algorithm is put forward to overcome the quantization problem. The algorithm uses a number of cost functions, and stages, equal to the number of incident signals. At each stage a new cost function is evaluated in a small number of "special" directions, known as characteristic points. For an N-element array the characteristic points, which can be pre-calculated from the array manifold curvatures, partition the array manifold into N-1 regions. By using a simple gradient algorithm, only a small area of one of these regions is searched at each stage, demonstrating the potential benefits of the proposed approach.