Abstract: The precise time step integration method proposed forlinear time-invariant homogeneous dynamic system can give precise numericalresults approaching to the exact solution at the integration points.However, it is more or less difficult when the algorithm is used to thenon-homogeneous dynamic systems due to the inverse matrix calculation andthe simulation accuracy of the applied loading. By combining the Gaussquadrature method and state space theory with the calculation technique ofmatrix exponential function in the precise time step integration method, anew precise time step integration method (that is renewal precise time stepintegration method) is proposed. The new method avoids the inverse matrixcalculation and the simulation of the applied loading and improves thecomputing efficiency. In particular, the method is independent to thequality of the matrix \pmb H. If the matrix \pmb H is singular or nearly singular,the advantage of the method is remarkable. The proposed method in this paperis a unconditionally stable algorithm having an arbitrary order of accuracy.Numerical examples are given to demonstrate the validity and efficiency ofthe algorithm.