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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:

- Transform the right hand side vector .
- Apply the (unpreconditioned) iterative method, replacing the coefficient matrix by ; call the resulting solution .
- Compute .

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.