Previous: Cost trade-off
Up: The why and how
Previous Page: Cost trade-off
Next Page: Jacobi Preconditioning
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:
- Transform the right hand side vector
.
- Apply the (unpreconditioned) iterative method, replacing
the coefficient matrix
by 
;
call the resulting solution 
.
- Compute
.
The important theoretical point is that the transformed
coefficient matrix 
preserves some theoretical properties of
and 
: if is symmetric and positive definite and
, then the transformed coefficient matrix
is again symmetric and positive definite.
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.