A decomposition multi-step non-iterative method for the numerical integration of short-term and long-term dynamics of power systems is given. This method is derived, on the one hand, on the basis of the triangular block alternative semilinear structural property, where it is to be noted that the triangular block alternative linear dynamical system is a special case of triangular block alternative semilinear dynamical system and, on the other hand, on the basis of representing the vector function of the time variable with a special form of interpolation polynomial. The method can be applied not only for the numerical integration of short-term and long-term dynamics of power systems, but also for all other cases where the model of the dynamical system can be represented in triangular block alternative semilinear form. Numerical stability, accuracy and computation speed of this method make suitable for many applications, such as the dynamic security analysis of power systems.