The PARAFAC method is an approach to decompose multidimensional arrays into component matrices for a given number of components. The most common way for calculating the decomposition is the alternating least squares method (ALS). Many other algorithms are modifications of ALS, including algorithms utilizing line search, enhanced line search or Tikhonov regularization. These algorithms and a new one, combining line search and Tikhonov regularization, are discussed in this paper. It is demonstrated that the new algorithm combines fast computation and successful handling of ill-conditioned problems, like given in the case of bottlenecks and swamps. Another point discussed is the application of compression for improving the algorithms considered. 2014 Elsevier B.V.