Copyright © 2015, Kent State University. We study the symmetric outer product for tensors. Specifically, we look at decompositions of a fully (partially) symmetric tensor into a sum of rank-one fully (partially) symmetric tensors. We present an iterative technique for third-order partially symmetric tensors and fourth-order fully and partially symmetric tensors. We include several numerical examples which indicate faster convergence for the new algorithms than for the standard method of alternating least squares.