This paper models an active intelligent reflecting surface (IRS) -assisted wireless communication network, which has the ability to adjust power between BS and IRS. We aim to maximize the signal-to-noise ratio of user by jointly designing power allocation (PA) factor, active IRS phase shift matrix, and beamforming vector of BS, subject to a total power constraint. To tackle this non-convex problem, we solve this problem by alternately optimizing these variables. Firstly, the PA factor is designed via polynomial regression method. Next, BS beamforming vector and IRS phase shift matrix are obtained by Dinkelbach's transform and successive convex approximation methods. To reduce the high computational complexity of the above proposed algorithm, we maximize achievable rate (AR) and use closed-form fractional programming method to transform the original problem into an equivalent form. Then, we address this problem by iteratively optimizing auxiliary variables, BS and IRS beamformings. Simulation results show that the proposed algorithms can effectively improve the AR performance compared to fixed PA strategies, aided by passive IRS, and without IRS.