Phase retrieval is the problem of recovering a vector when the only the magnitudes of a set of linear functionals is known, while phase information is unavailable. We present an algorithm for finite dimensional noisy phase retrieval and an extension of the algorithm to phase retrieval of a sparse vector, both with explicit error bounds that are linear in the noise-to-signal ratio. In the case of an s-sparse vector in an N dimensional space, we show that a number of measurements on the order of s log(N/s) is sufficient to obtain a high probability of recovery.