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This paper is concerned with the design of second-order algorithms for an equalizer in a training or a tracking mode. The algorithms govern the iterative adjustment of the equalizer parameters for the minimization of the mean-squared error. On the basis of estimated bounds for the eigenvalues of the signal plus noise correlation matrix, an optimal second-order algorithm is derived. The resultant convergence is considerably faster than the commonly used first-order fixed-size gradient-search procedure. The variance of the optimal algorithm is shown to have a slightly larger bound than the present first-order fixed-step algorithm. However, a computer simulation for an input signal-to-noise ratio of 30 dB shows that for large intersymbol interference the improvement in the convergence of the mean more than compensates for the small increase in variance. For moderate intersymbol interference the simulation shows no variance increase. Suboptimum second-order algorithms with smaller improvement in the convergence rate and smaller increase in the variance bound are also considered. The results indicate that, on the average, the new algorithms lead to faster tracking of changes in the channel characteristics and thereby result in a smaller error rate.