As described above, a simple fixed point iteration scheme for the solution of the rate equations will converge much too slowly to be useful for most cases of practical interest. Therefore, we use an extension of the operator splitting idea for the solution of the rate equations.

We rewrite the rate equations in the form of an ``operator equation.'' This equation is then used to introduce an ``approximate rate operator'' in analogy to the approximate -operator which can then be used to iteratively solve the rate and statistical equations by an operator splitting method, details of the approach are given in [2].

We introduce first the ``rate operator'' [*R*_{ij}] for upward transitions
in analogy to the -operator. [*R*_{ij}] is defined so that

Using the rate operator, we can write the rate equations in the form

In analogy to the operator splitting method discusses above, we split the rate
operator, by writing (analog for the downward radiative rates), where is the
``approximate rate-operator''. We then rewrite the rate *R*_{ij} as

If we insert Eq. 26 into Eq. 16, we obtain the following
system for the new population densities:

Due to its construction, the [R_ij^*][*R*_{ij}^{*}]-operator contains information
about the influence of a particular level on *all* radiative
transitions. Therefore, we are able to treat the complete multi-level
non-LTE radiative transfer problem including active continua and
overlapping lines. The -operator, at the same time,
gives us information about the strength of the coupling of a radiative
transition to all levels that are considered. This information may be
used to include or neglect certain couplings *dynamically* during
the iterative solution of Eq. 27. Furthermore, we have not
yet specified either a method for the formal solution of the radiative
transfer equation or a method for the construction of the approximate
-operator (and, correspondingly, the -operator). We proceed
by considering rapidly expanding spherically symmetric media and use
the tri-diagonal ALO given by Hauschildt [1]. However, any
method for the formal solution of the radiative transfer equation and
the construction of the ALO may be used, including multi-dimensional
and/or time dependent methods.