By Joyner D., Kreminski R., Turisco J.

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Let y 0 2 R2d , M and N > 0 be fixed. y n /, n 0. Assume that for all n 0, we have ky n k Ä M . y / ˇ ˇ H N y0 ˇ Ä C n N C1 where C depends on M and N . y n / H y 0 ˇ Ä c ; for n Ä for some constant c depending on N and M . 26) 38 II Finite dimensional backward error analysis Proof. y n / Ä 2M . y n /ˇ Ä B ˇH y @ sup kyk Ä 3M Ä 0 ÄC 0 HN . y n / N C1 for some constant C depending on M and N (as 0 does). 25) by induction. y n / H N y 0 ˇˇ ˇ ˇ ˇ ˇ H y0 ˇ C ˇH N y 0 ÄC Cn N C1 Ä 2C for n Ä N ; for some constant C .

Q/ respectively, which makes splitting methods very easy and cheap to implement. 1). e. the mapping In the following, we denote by 'H 0 t 2d 2d 'H W R ! 4) t is defined for all t > 0, all y 2 R2d , and is smooth. We will always assume that 'H This is guaranteed for example when H is smooth and when the solution remains bounded. 2) is associated the symplectic form ! W R2d 2d R ! R such that !. 1. A matrix A of size 2d is symplectic if it satisfies AT JA D J: A differentiable nonlinear mapping ' W R2d !

The derivative of a given monomial zj D zj1 zji with respect to zj vanishes except if j j . Assume for instance that j D ji . 20) after summing in i D 2; : : : ; k. `1s ; C/. 1 t/z/ dt ˇˇ jzq jj j ˇ @zj @zj 0 @zj @zq yq j: q2Z But we have, for fixed j D . `1s ; `1s /. The fact that the Hamiltonian P and the vector field XP are C 1 can be verified using similar calculations. Assume now that P and Q are homogeneous polynomials of degrees k and ` respectively and with coefficients ak , k 2 Ik and b` , ` 2 I` .