n this section we document a negative
result. We make a natural attempt to switch to a symmetrical problem and find
that such transformation does not bring noticeable benefit.
We start from the
equation
of the section (
Diagonal
preconditioner
) and multiply it with the operator
.
We obtain a system with symmetrical
matrix:
We aim for the matrixes
and
to be as small as possible. The symmetrization step took us in opposite
direction.
Matrix
for
.

We note that we cannot apply the diagonal preconditioner of the section
(
Diagonal preconditioner
)
because such operation looses the symmetry. Hence, we attempt the following
transformation:
for some new diagonal matrix
that we select from minimization of Frobenius norm. We
calculate
and we select
by minimizing
.
Observe
that
then
We seek the local
minimum:
We arrived to a system of linear
equations:
The symmetry is the only good feature of this system.
Matrix
for
.

The numerical experiment in the script blackDp.py shows that the condition
number of
for
is 1883. The good thing is we do not need
to be exact. However, given results of the nonsymmetric line of calculation,
this is not worth pursuing further.
