e noted in the section
(
Diagonal preconditioner
) that
the properties of the matrix
of the
system
improve if we replace
with powers of
.
The calculations of the section
(
Analytical
preconditioner derived from asymptotic decomposition in time
) hint on how
we would use such observation.
We multiply the equation
with
:
and
obtain
We multiply with
:
and
continue
Numerical experimentation in the script blackDp.py show that the condition
number
for matrix
computed from
improves as
follows
The maximal singular values
and Frobenius norms of
are
This procedure works as a substitution for inversion.
