Calculating the phase shifts assigned to the elements of a phased array antenna such that the resulting beam is shaped to serve a desired area of operation (AOO) has historically been computationally burdensome and often required expert intervention. To maximize computational speed, synthesis of shaped phased-array antenna beams is performed by linearizing the antenna pattern equation and then iteratively performing a mini-norm solution at each step until a solution is reached. In particular, this approach is performed in such a manner that eliminates the need for a pre-computed target and is also performed such that at each iteration the change in an element's phase is adapted to remain within a threshold range. As a result, phased array beam patterns may be synthesized and applied to phased-array antennas so as to allow real time tracking of AOOs on the Earth from Low and Medium Earth Orbit satellites.