A note on symplectic and symmetric ARKN methods

For dealing with the oscillatory separable Hamiltonian system of the form q″(t)+ω2q(t)=f(q(t)) with ω(>0), the main frequency of the system, the symplecticity conditions of ARKN methods and the existence of symmetric ARKN methods are investigated by Shi et al. [W. Shi, X. Wu, Comput. Phys. Comm. 183 (2012) 1250]. However, the symplecticity conditions in that paper are not sufficient, one of the conditions should be made stronger to guarantee the symplecticity of the methods. For this reason, we will reanalyze and present more precisely the symplecticity conditions for the ARKN methods in this note.