  We present a simplified computational rule for the back-propagation formulas
for artificial neural networks. In this work, we provide a generic two-step
rule for the back-propagation algorithm in matrix notation. Moreover, this rule
incorporates both the forward and backward phases of the computations involved
in the learning process. Specifically, this recursive computing rule permits
the propagation of the changes to all synaptic weights in the network, layer by
layer, efficiently. In particular, we use this rule to compute both the up and
down partial derivatives of the cost function of all the connections feeding
into the output layer.
