  A detailed exposition of highly efficient and accurate method for the
propagation of the time-dependent Schr\"odinger equation is presented. The
method is readily generalized to solve an arbitrary set of ODE's. The
propagation is based on a global approach, in which large time-intervals are
treated as a whole, replacing the local considerations of the common
propagators. The new method is suitable for various classes of problems,
including problems with a time-dependent Hamiltonian, nonlinear problems,
non-Hermitian problems and problems with an inhomogeneous source term. In this
paper, a thorough presentation of the basic principles of the propagator is
given. We give also a special emphasis on the details of the numerical
implementation of the method. For the first time, we present the application
for a non-Hermitian problem by a numerical example of a one-dimensional atom
under the influence of an intense laser field. The efficiency of the method is
demonstrated by a comparison with the common Runge-Kutta approach.
