Abstract: The structural dynamics equations were converted to the state equations in which the displacement and velocity response were taken as the state variables. In order to solve the state equations, the perturbation method was used, and a new series form of analytical solutions was presented. At the same time, the corresponding iterated computation formats and steps for the dynamics equations were established. The algorithm needs only repetitious matrix-vector multiplication and vector summation without inversion of matrix and calculation of exponential matrix Thusly, the computation stability and efficiency are very high. The items of the series solution and the accuracy of the algorithm can be directly controlled by the tolerance parameter, and theoretically, the algorithm can easily achieve arbitrary-order accuracy, and be suitable for parallel computing and compression storage. Generally, the algorithm combines the high-efficiency of the linear acceleration methods and the high-precision of precise integration methods. This method can be used for calculating the large sparse linear dynamic equations of engineering structures. At last, a model numerical example was given to demonstrate the validity and efficiency of the method.