In this paper, the displacement and velocity response of a structure are taken as the state variable. In order to solve the state equation, the Lyapunov’s artificial small parameter method is used, and a new series solution for the time-discrete form of the equation is presented. Hornor’s scheme is applied to the calculation of the series solution, and the computational efficiency and stability are therefore greatly improved. The corresponding computation scheme and procedure for the proposed algorithm are also presented. The algorithm is only composed of repetitious matrix-vector multiplication and vector summation and the calculations of the inverse matrix H
61485;1 and exponential matrix are excluded, which results in high computational efficiency and stability. The accuracy of the algorithm is only controlled by the parameters related to the series, thusly the algorithm can easily achieve arbitrary-order accuracy in theory, and is suitable for parallel computing and compression storage. The proposed method can be used for seismic calculations of engineering structures. Two numerical examples are given to demonstrate the validity and efficiency of the method.