A survey of the precise integration method

Abstract

For the initial value and two-point boundary value problems of linear ordinary differential equation, the Precise Integration Method (PIM) gives exact solutions in the computer accuracy sense. In this paper, the basic idea and the further development of PIM are surveyed. For the initial value problems, the basic idea of PIM, the methods for integrating the nonhomogeneous term, the methods for large scale problems and the application of PIM in time-varying and nonlinear system are surveyed. For two-point boundary value problems, the basic idea and the fundamental formula of PIM are given and the application of the PIM for two-point boundary value problems in many fields are surveyed. Finally, the relationship between the PIM for the initial value and two-point boundary value problems is discussed, which gives a new angle for understanding and application of PIM.