THIS PAPER IS ELIGIBLE FOR THE STUDENT PAPER AWARD. We design an algorithm from the perspective of information theory to calculate the capacity region of the Gaussian vector broadcast channel with private messages. For a continuous channel, a classic method to approximately calculate its capacity is to apply the Blahut-Arimoto algorithm after discretization. In this work, we decouple the coupled variables in the original problem by exploiting the property that a Gaussian distribution is uniquely determined by its mean and variance. Furthermore, we derive an equivalent form of the objective function that facilitates simpler projection operations, and thus develop a Gaussian Blahut-Arimoto algorithm without discretization.