For later discussion, let us first introduce the
operator 
defined by
The adjoint of with respect to the standard inner products
in 
and 
is the operator
The main step at each iteration of our algorithms is the
computation of the search direction 
from the symmetrized Newton equation (with respect to
an invertible matrix P which is usually chosen as a function of the
current iterate X,Z) given below.
where 
and 
is the
centering parameter. Here
is the symmetrization operator defined by
and 
and 
are the linear operators
where 
denotes the linear operator defined by
Assuming that 
, we compute the search direction via
a Schur complement equation as follows
(the reader is referred to [2] and [8] for details).
First compute 
from the Schur complement equation
where
Then compute 
and 
from the equations
If 
, solving (9) by a direct
method is overwhelmingly expensive;
in this case, the user should consider using an implementation that
solves (9) by an iterative method
such as CG or QMR.
In our package, we assume that and
(9) is solved by a direct method such
as LU or Cholesky decomposition.
