e multiply the
equation
of the summary
(
Reduction
to system of linear algebraic equations for q=1
) with a diagonal matrix
:
and transform the
matrix
or
We would like to choose the matrix
so that the Frobenius norm of the
matrix
would be minimal.
We
calculate
thus
We observe that
is a positive quadratic function of
,
thus the minimum is located where the derivatives
vanish. We
differentiate
and
transform
Calculation of the script blackDp.py in the directory
OTSProject/python/wavelet2 shows that such preconditioner is very effective.
For the parameters, set in the section
(
Example Black equation
parameters
) and
,
the matrix
has the following
metrics:
The matrix
has the following
metrics:
If
then the matrix
has the following
metrics:
If
then the matrix
has the following
metrics:
Note that even for
the spectral radius stays below
.
In addition, the Frobenius norm
decreases if
increases and the maximal singular value drops below 1 for reasonably large
.
