The Step-Transition Operators for Multi-Step Methods of ODE's (Journal of Computational Mathematics, 1998)
In this paper, we propose a new definition of symplectic multistep methods. This definition differs from the old ones in that it is given via the one step method defined directly on $M$ which is corresponding to the $m$ step scheme defined on $M$ while the old definitions are given out by defining a corresponding one step method on $M× M × … × M=M^m$ with a set of new variables. The new definition gives out a step-transition operator $g: M\longrightarrow M$. Under our new definition, the Leap-frog method is symplectic only for linear Hamiltonian systems. The transition operator $g$ will be constructed via continued fractions and rational approximations.
Keywords: Multi-step methods; Explike and loglike function; Fractional and rational approximation; Simplecticity of LMM; Nonexistence of SLMM.