We solve the nonlinear system 
by
constructing a sequence of improved solutions 
with the
residuals 
. Improved means that the
residuals' norms decrease with each update (
). The nonlinear system
is assumed to be solved when
the residual norm 
is sufficiently small
(3.5-9).
The termination criterion 
(Newton accuracy) has to be consistent with
the magnitude of the terms in 
. For simulators where the equations
are known a priori, e.g. for device simulators, it is possible to
find an appropriate scaling from physical considerations. In PROMIS we do not make any restrictions to specific equations or physical
problems and, therefore, scaling is based purely on numerical
considerations.
We have incorporated a local scaling for the system 
. For
each equation 
at each grid point 
we apply a local scaling factor
(3.5-10) derived from (3.5-7).
The variables 
themselves remain unscaled and, therefore, in addition
to the right hand side 
, the rows of the Jacobian matrix
are scaled with the identical factors (3.5-12).
A consequence of the local scaling is that all equations are solved to the same degree of accuracy at all grid points independent of the magnitude of the involved physical quantities. In the further text, we omit the explicit indication of the scaled equations and use the same symbols irrespective on whether they are scaled or not.
