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The above transformation of the linear system
is not what is used in practice. A more correct way of introducing the
preconditioner would be to split the preconditioner as
and
to transform the system as
The matrices and
are called the left-
and right preconditioners
, respectively.
An iterative method can be preconditioned according to the following scheme:
Since symmetry and definiteness are crucial to the success of some
iterative methods, this transformation is to be preferred over
, which is not guaranteed to be either symmetric or definite,
even if
and
are.
It is a remarkable property of many iterative methods that the splitting
of is in practice not needed.
By rewriting the steps of the method (see for
instance Axelsson and Barker ([14],pgs. 16,29) or Golub and Van Loan [108], 10.3) it
is usually possible to reintroduce a computational step
that is, a step that applies the preconditioner in its entirety.