The systems of nonlinear equations
are solved by a combination of a damped
Newton iteration [16, 15]
and the hierarchical basis multigrid iteration
[12, 11, 20, 5, 6, 38, 39].
When the problem has 
dependence, the bordered system of
equations is solved by block Gaussian elimination; this requires the
solution of two sets of equations using the multigrid iteration.
Suppose the system to be solved has the form :
Here the operator G represents the finite element equations
of order NVF, and
N the normalizing equation used in the continuation process;
is the steplength.
At any given step of the Newton process,
the system to be solved has the form
where 
is a vector of length NVF and 
is
a scalar.
The solution is constructed by solving
The coefficient 
, provided 
;
otherwise.
Thus the right hand side 
has the appearance
of a residual, and w may be viewed as an incremental update.
In particular, the vector 
is proportional
to the next tangent vector at convergence. The tentative tangent
is thus easily updated at every Newton step,
along with u, , 
, 
and 
,
and is available for regularizing the right hand side
for w on the next Newton step.
Since all linear systems involving 
and 
are solved iteratively, solving for corrections
using a zero initial guess is a good strategy for minimizing the
number of iterations.
The block elimination process is embedded in an overall
damped Newton process [14, 15]
given in Figure 
.
Here 
is the k-th Newton iterate,
, and
.
The norm 
is given by
where c is a scaling parameter (SCALE in the RP array) chosen to balance the two terms appropriately.
The scalar 
is the damping parameter.
When the sufficient decrease criterion is not satisfied on line N4,
and must be reduced, the next value is found through application
of one step of a guarded secant/bisection algorithm
to the one-dimensional minimization problem
If sufficient decrease is achieved, the current is used to
predict 
; this formula is designed to force rapid increase of
to one when 
becomes small as superlinear
convergence occurs, and at the same time, provide a reasonable first
guess in the early stages of the Newton iteration, when damping is most
important.
The same damping strategy is applied when
there is no dependence, with the obvious redefinition of
, 
, and 
.
A maximum of ITMAX
damped Newton iterations are allowed, where
ITMAX is a user specified integer. PLTMG reports the actual number
of Newton iterations used on the most recent call in the parameter
ITNUM, and the number of evaluations of 
as IEVALS;
, since more than one function evaluation may be used
in each line search.
