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The FastICA or fixed-point algorithm is one of the most successful algorithms for linear independent component analysis (ICA) in terms of accuracy and computational complexity. Two versions of the algorithm are available in literature and software: a one-unit (deflation) algorithm and a symmetric algorithm. The main result of this paper are analytic closed-form expressions that characterize the separating ability of both versions of the algorithm in a local sense, assuming a "good" initialization of the algorithms and long data records. Based on the analysis, it is possible to combine the advantages of the symmetric and one-unit version algorithms and predict their performance. To validate the analysis, a simple check of saddle points of the cost function is proposed that allows to find a global minimum of the cost function in almost 100% simulation runs. Second, the Crame´r-Rao lower bound for linear ICA is derived as an algorithm independent limit of the achievable separation quality. The FastICA algorithm is shown to approach this limit in certain scenarios. Extensive computer simulations supporting the theoretical findings are included.