(untitled) (Journal of Computational Mathematics, 2021)
We prove that any linear multi-step method G_1^τ of the form ^m_k=0α _k Z_k=τ ^m_k=0β _k J^-1 abla H(Z_k) with odd order u (u3) cannot be conjugate to a symplectic method G_2^τ of order w (wu) via any generalized linear multi-step method G_3^τ of the form ^m_k=0 α_k Z_k = τ^m_k=0 β_k J^-1 abla H(^m_l=0γ_klZ_l). We also give a necessary condition for this kind of generalized linear multi-step methods to be conjugate-symplectic. We also demonstrate that these results can be easily extended to the case when G_3^τ is a more general operator.