Up: The why and how
Next Page: Jacobi Preconditioning

## Theoretical prerequisites on preconditioners

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:

1. Transform the right hand side vector .
2. Apply the (unpreconditioned) iterative method, replacing the coefficient matrix by ; call the resulting solution .
3. 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.