The four equations provide us with a way of computing the gradient of the cost function. Let us write this out in the form of an algorithm.
First, we set up the input data \( \hat{x} \) and the activations \( \hat{z}_1 \) of the input layer and compute the activation function and the pertinent outputs \( \hat{a}^1 \).
Secondly, we perform then the feed forward till we reach the output layer and compute all \( \hat{z}_l \) of the input layer and compute the activation function and the pertinent outputs \( \hat{a}^l \) for \( l=1,2,3,\dots,L \).
Notation: The first hidden layer has \( l=1 \) as label and the final output layer has \( l=L \).