  In the present article, we review a continual effort on generalization of the
Trotter formula to higher-order exponential product formulas. The exponential
product formula is a good and useful approximant, particularly because it
conserves important symmetries of the system dynamics. We focuse on two
algorithms of constructing higher-order exponential product formulas. The first
is the fractal decomposition, where we construct higher-order formulas
recursively. The second is to make use of the quantum analysis, where we
compute higher-order correction terms directly. As interludes, we also have
described the decomposition of symplectic integrators, the approximation of
time-ordered exponentials, and the perturbational composition.
