ccording to the proposition
(
Convergence of conjugate
gradient method
), the procedure of the summary
(
Conjugate gradients
) converges
significantly better when
and
are close. For this reason one may attempt to
consider
instead of
for some matrix
that almost inverts
.
The matrix
does not have to be symmetric or positive definite in
.
However, it is self-adjoint and positive definite with respect to
:
Therefore, it has all the necessary spectral properties and we can still apply
the procedure (
Conjugate gradients
).
Another possibility is to try
factorization
Such decomposition always exists (but not unique) if
is symmetric positive definite. We
have
We make the
change
multiply by
and arrive
to
Note that (by similar calculation)
The matrix
is symmetric positive-definite in
.
The procedure (
Conjugate gradients
) then
can be adapted for the equation
with usual tricks to keep down the number of matrix multiplications.
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