Abstract : Several modifications have been proposed to speed up the Alternating Least Squares (ALS) method of fitting the Parafac model. The most widely used is Line Search, which extrapolates from linear trends in the parameter changes over prior iterations to estimate the parameter values that would be obtained after many additional ALS iterations. We propose some extensions of this approach that incorporate a more sophisticated extrapolation, using information on nonlinear trends in the parameters and changing all the parameter sets simultaneously. The new method, called 'Enhanced Line Search', can be implemented at different levels of complexity, depending on how many different extrapolation parameters (for different modes) are jointly optimized during each iteration. We report some tests of the simplest parameter version, using simulated data. The performance of this lowest-level of ELS depends on the nature of the convergence difficulty. It significantly outperforms standard LS when there is a ``convergence bottleneck'', a situation where some modes have almost collinear factors but others do not, but is somewhat less effective in classic ``swamp'' situations where factors are highly collinear in all modes. This is illustrated by examples. To demonstrate how ELS can be adapted to different N-way decompositions, we also apply it to a four-way array to perform a Blind Identification of an Under-Determined Mixture (UDM). Since analysis of this dataset happens to involve a serious convergence ``bottleneck'' (collinear factors in two of the four modes), it provides another example of a situation in which ELS dramatically outperforms standard Line Search.