I'm learning about iterative methods to solve the system $Au = f$, and I came across the following in my book:
To construct an iterative scheme, we assume that a non-singular matrix $M$ exists and define the matrix $N$ as $N = M - A$. We can then write $$A = M - N$$ The linear system $Au = f$ can then be written as $Mu = Nu + f$. By multiplying to the left and right with $M^{-1}$ we can define an iterative scheme $$\begin{split}u^{k+1} &= M^{-1}Nu^k + M^{-1}f\\&=M^{-1}(M-A)u^k + M^{-1}f\\&=u^k + M^{-1}(f - Au^k)\\&=u^k + M^{-1}r^k\end{split}$$
Question: This all makes a lot of sense to me, except for one thing; why is it allowed to write $u^{k+1} = M^{-1}Nu^k + M^{-1}f$ with the superscripts $k+1$ and $k$ over $u$? The construction of an iterative method goes from $Mu = Nu + f$ to $u^{k+1} = M^{-1}Nu^k + M^{-1}f$ and I don't understand how.
Thanks in advance!
The concept of iterative methods is to use the approximations obtained in an earlier step to generate new approximations using the same process over and over again. So in this case, from the equation, $u=M^{-1}Nu+M^{-1}f$, the solution $u$ is initially assigned some value say $u^0$. This is then fed in the RHS of the above equation to obtain another approximation $u^1$ of $u$ as $$u^1=M^{-1}Nu^0+M^{-1}f.$$ Then $$u^2=M^{-1}Nu^1+M^{-1}f$$ and so on, which is expressed as $$u^{k+1}=M^{-1}Nu^k+M^{-1}f.$$