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The polynomial acceleration method for time history analysis is presented, in which accelerations between several equal neighboring time steps were assumed to be a polynomial function of time interval. With a higher order polynomial used, a higher accuracy can be obtained, but the stability field of the method becomes narrower. When stability field and computational accuracy are taken into account at the same time, the quadratic acceleration method is superior to linear and cubic acceleration methods in choosing the maximum acceptable time step size. It is also shown that the quadratic acceleration method has desirable arithmetic damp, amplitude decay rate and period elongation rate, though its conditional stability restricts its application in stiff structures.