A note on symplecticity of step-transition mappings for multi-step methods

We prove that for a linear multi-step method ∑ k = 0 m α k Z k = τ ∑ k = 0 m β k f ( Z k ) , even though the mappings Z 0 → Z 1 , … , Z m - 2 → Z m - 1 are chosen to be symplectic, Z m - 1 → Z m will be non-symplectic. Similarly, there is an interesting result for a sort of general linear methods.