Baker-Hausdorff theorem

Note that when a symplectic integration method is used, we have, from the discussion in Chap. 3, a perturbed energy function and, moreover, from Theorem 3.1, the error in energy H is OQf), nonetheless the perturbations are large for a large stepsize h. In the Shadow Hybrid Monte-Carlo (SHMC) method [2, 3, 188], the accept-reject test is based on the modified Hamiltonian Hh (see Chap. 3), derived from the Baker-Campbell-Hausdorff expansion. SHMC can improve efficiency by decreasing the rejection rate. 

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