Given a linear system $A\textbf{x}=\textbf{b}$, we can express it into the easier-to-solve right-preconditioned form:

$$ AM^{-1}\textbf{y}=\textbf{b}, \quad \textbf{y}= M^{-1}\textbf{x} $$

On the other hand, the left-preconditioned system is:

$$ M^{-1}A\textbf{x}=M^{-1}\textbf{b} $$

From the literature we know it's easy to show that the left-preconditioned linear system above is equivalent to a fixed-point linear iteration as:

$$ \textbf{x}^{(k+1)}=(I-M^{-1}A)\textbf{x}^{(k)}+M^{-1}\textbf{b} $$

Or equivalently:

$$ \textbf{x}^{(k+1)}=\textbf{x}^{(k)} + M^{-1}\textbf{r}^{(k)} $$

My question is: how can I get an update scheme like those starting from the right-preconditioning formulation?


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