How to derive the error $e^{n}=M^{n}e^{0}$ from iteration $x^{n+1} = Mx^{n} + f$?

Question

$x^{n+1} = Mx^{n} + f$ is fixed-point iteration for solving the equation $x = Mx + f$, i.e., $(I-M)x = f$.

The error $e^{n} = x - x^{n}$

How does one get $e^{n}=M^{n}e^{0}$?

Answer 1

Answer:

$$e^0 = x^1 - x^0$$ $$e^1 = x^2 - x^1$$ $$e^2 = x^3 - x^2$$

$$x^1 = Mx^0+f$$ $$e^0 = x^0(M-1) +f$$ $$x^2 = M(Mx^0+f) - x^0$$ $$x^2 = M^2x^0+Mf+f$$ $$e^1 = Me^0$$

$$x^3 = M^3x^0 +M^2f+Mf+f$$

$$e^2 = x^3-x^2$$

$$e^2 = M^2(x^0(M-1)+f)$$

$$e^2 = M^2e^0$$

By induction:

$$e^n = x^{n+1} - x^n$$

$$e^n = M^ne^0$$