The formal solution of the SSRTE is performed along the characteristic
rays on a mesh , of discrete shells using the
short-characteristic (SC) method of Olson and Kunasz [24] with
piece-wise parabolic or linear interpolation. The characteristic rays are
*curved* in the case of the SSRTE and have to be calculated before the
solution of the radiative transfer equation proceeds (see [1]
for details). Improvements in this method [1,25] include
an improved angle integration using generalized Simpson-quadrature and
a generalization of the approximate -operator to an arbitrary number
of bands below and above the main diagonal (up to the full -operator).

We describe here the general procedure of calculating the with
*arbitrary* bandwidth, up to the full -operator, for the
SC method in spherical symmetry [25]. Although we consider the
SSRTE as given in the previous section, the same procedure applies for
radiative transfer problems in static media or in (static or moving)
media with plane-parallel symmetry. The specialization of the formulae
given in this section is straightforward.

The formal solution along a characteristic of the SSRTE (hereafter, a
``ray'') is done using a polynomial interpolation of the source
function, *S*, along the ray. For reasons of numerical stability, we
use linear or quadratic interpolation of *S* along each ray,
although this is not required by the method. This leads to the
following expressions for the specific intensity along a
ray (cf. Ref. [24] for a derivation of the formulae):

We describe the construction of for arbitrary bandwidth using
the example of a characteristic that is tangential to an arbitrary shell:
Ray *k* is the ray that is
tangent to shell *k*+1
The intersection points (including the point of
tangency) are labeled from left to right, the direction in which the
formal solution proceeds. Ray *k* has 2*k*+1 points of
intersection with discrete shells . To compute row *j* of
the discrete -operator (or -matrix), ,we sequentially label the intersection points of the ray *k* with the
shell *i*, and define auxiliary
quantities and as follows:

Using the and , we can now write the
-Matrix as