Prior to the recent development of symplectic integrators, the time-stepping operator was routinely decomposed into a sum of products of and in the study of hyperbolic partial differential equations. In the context of solving Hamiltonian dynamics, we show that such a decomposition gives rise to both even- and odd-order Runge–Kutta and Nyström integrators. By the use of Suzuki’sforward-time derivative operator to enforce the time-ordered exponential, we show that the same decomposition can be used to solve nonautonomous equations. In particular, odd-order algorithms are derived on the basis of a highly nontrivial time-asymmetric kernel. Such an operator approach provides a general and unified basis for understanding structure nonpreserving algorithms and is especially useful in deriving very high-order algorithms via analytical extrapolations. In this work algorithms up to 100th order are tested by integrating the ground-state wave function of the hydrogen atom. For such a singular Coulomb problem, the multi-product expansion shows uniform convergence and is free of poles usually associated with structure-preserving methods. Other examples are also discussed.