  Prior to the recent development of symplectic integrators, the time-stepping
operator $\e^{h(A+B)}$ was routinely decomposed into a sum of products of
$\e^{h A}$ and $\e^{hB}$ in the study of hyperbolic partial differential
equations. In the context of solving Hamiltonian dynamics, we show that such a
decomposition give rises to both {\it even} and {\it odd} order Runge-Kutta and
Nystr\"om integrators. By use of Suzuki's forward-time derivative operator to
enforce the time-ordered exponential, we show that the same decomposition can
be used to solve non-autonomous equations. In particular, odd order algorithms
are derived on the basis of a highly non-trivial {\it time-asymmetric} kernel.
Such an operator approach provides a general and unified basis for
understanding structure non-preserving algorithms and is especially useful in
deriving very high-order algorithms via {\it analytical} extrapolations. In
this work, algorithms up to the 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 showed uniform convergence and is free of
poles usually associated with structure-preserving methods. Other examples are
also discussed.
