  We show that the method of factorizing the evolution operator to fourth order
with purely positive coefficients, in conjunction with Suzuki's method of
implementing time-ordering of operators, produces a new class of powerful
algorithms for solving the Schroedinger equation with time-dependent
potentials. When applied to the Walker-Preston model of a diatomic molecule in
a strong laser field, these algorithms can have fourth order error coefficients
that are three orders of magnitude smaller than the Forest-Ruth algorithm using
the same number of Fast Fourier Transforms. When compared to the second order
split-operator method, some of these algorithms can achieve comparable
convergent accuracy at step sizes 50 times as large. Morever, we show that
these algorithms belong to a one-parameter family of algorithms, and that the
parameter can be further optimized for specific applications.
