# Right-preconditioning and fixed point linear iterations

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

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

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?

Starting with

$$AM^{-1} \mathbf{y} = \mathbf{b}$$, where $$\mathbf{y} \equiv M \mathbf{x}$$,

we can manipulate to

$$\mathbf{y} - (I-AM^{-1})\mathbf{y} = \mathbf{b}$$

Replacing one instance of $$\mathbf{y}$$ by $$\mathbf{y}^k$$ and the other by $$\mathbf{y}^{k+1}$$ yields the update equation

$$\mathbf{y}^{k+1} = (I-AM^{-1})\mathbf{y}^k + \mathbf{b}$$

which can also be written as

$$\mathbf{y}^{k+1} = \mathbf{y}^k + (\mathbf{b}-AM^{-1}\mathbf{y}^k)$$