Iterative method

Remarks

A stationary method is a fixed-point iteration for $\Phi$. For a linear system $A x = b$ with $A$ invertible, a stationary linear method has the form $x_{k+1} = G x_k + c$. It is consistent if $x^* = A^{-1} b$ satisfies $x^* = G x^* + c$. Then the error $e_k = x_k - x^*$ satisfies $e_{k+1} = G e_k$. So the method converges for every $x_0$ if and only if the spectral radius of $G$ is less than $1$. Example: $x_{k+1} = x_k / 2 + 1$ on $\R$ is stationary and consistent for $x / 2 = 1$, with $x^* = 2$. Its error halves at each step, so it converges for every $x_0$. Non-example: $x_{k+1} = 2 x_k - 2$ has the fixed point $2$, but the error doubles at each step. It is not convergent for any $x_0 \ne 2$. Krylov subspace methods are nonstationary: each iterate depends on all earlier ones, or on several of them.