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In this paper, we propose a method for the exact computation of the Hessian matrix of the training error function for a multilayer perceptron network. The Hessian matrix is divided into small submatrices, which are calculated independently and then assembled. We developed a new pruning technique using the Hessian to estimate the error deviation due to the elimination of connections in the network. The method proposed is applied in the optimization of a loudspeaker's magnet problem consisting of seven design variables. The number of input variables is reduced while achieving the objective of the problem at an acceptable computational time.