The elements of the Hessian matrix consist of the second derivatives of the error measure with respect to the weights and thresholds in the network. They are needed in Bayesian estimation of network regularization parameters, for estimation of error bars on the network outputs, for network pruning algorithms, and for fast re-training of the network following a small change in the training data. In this paper we present an extended back-propagation algorithm which allows all elements of the Hessian matrix to be evaluated exactly for a feed-forward network of arbitrary topology. Software implementation of the algorithm is straightforward.