On precise integration method

The numerical integration for the ordinary differential equations is extremely important for applications. So far, almost all the numerical integration methods apply the finite difference approximation even for a time-invariant system.The precise integration method (PIM) solves the time step integration for the time-invariant system first. For such a problem, precise integration gives a highly precise numerical result, which approaches the full computer precision. After the integration of time-invariant system is solved, various approximate methods can be applied to other problems such as time-variant or nonlinear time integration.The numerical integration of two-point boundary value problems (TPBVP) is also very important in applications. Such as wave propagation, optimal control, structural mechanics, electro-magnetic wave guide problems, etc. The PIM can also be applied to solve the TPBVP. The TPBVP induced initial value problems such as the Lyapunov differential equation, the Riccati differential equation and the Kalman–Bucy filter equation, etc. can also be solved along the same way.In this paper, the essence of precise integration will be explained.