We introduce a novel fast algorithm for independent component analysis, which can be used for blind source separation and feature extraction. We show how a neural network learning rule can be transformed into a fixedpoint iteration, which provides an algorithm that is very simple, does not depend on any user-defined parameters, and is fast to converge to the most accurate solution allowed by the data. The algorithm finds, one at a time, all nongaussian independent components, regardless of their probability distributions. The computations can be performed in either batch mode or a semiadaptive manner. The convergence of the algorithm is rigorously proved, and the convergence speed is shown to be cubic. Some comparisons to gradient-based algorithms are made, showing that the new algorithm is usually 10 to 100 times faster, sometimes giving the solution in just a few iterations.