The above transformation of the linear system is often not what is used in practice. For instance, the matrix is not symmetric, so, even if and are, the conjugate gradients method is not immediately applicable to this system. The method as described in figure remedies this by employing the -inner product for orthogonalization of the residuals. The theory of the cg method is then applicable again.
All cg-type methods in this book, with the exception of QMR, have been derived with such a combination of preconditioned iteration matrix and correspondingly changed inner product.
Another way of deriving the preconditioned conjugate gradients method would be to split the preconditioner as and to transform the system as
If is symmetric and , it is obvious that we now have a method with a symmetric iteration matrix, hence the conjugate gradients method can be applied.
Remarkably, the splitting of is in practice not needed. By rewriting the steps of the method (see for instance Axelsson and Barker [pgs. 16,29]AxBa:febook or Golub and Van Loan [.3]GoVL:matcomp) it is usually possible to reintroduce a computational step
that is, a step that applies the preconditioner in its entirety.
There is a different approach to preconditioning, which is much easier to derive. Consider again the system.
The matrices and are called the left- and right preconditioners , respectively, and we can simply apply an unpreconditioned iterative method to this system. Only two additional actions before the iterative process and after are necessary.
Thus we arrive at the following schematic for deriving a left/right preconditioned iterative method from any of the symmetrically preconditioned methods in this book.
where is the final calculated solution.