For a system shown below
$$x_{k+1}=Gx_k+f$$
Where \(G\) is the iteration matrix. Convergence is guaranteed if the initial guess \(x_0\) if \(\rho(G)<1\) where the spectral radius \(\rho(G)\) is the largest eigenvalue of G. If the method converges, then the convergence factor is
$$\phi=\lim_{k\rightarrow\infty}\Big(\max_{x\neq x_0\in R_n}\frac{||x_k-x||}{||x_0-x||}\Big)^{\frac{1}{k}}=\rho(G)$$
The error \(||x_k-x||\) decreases by a factor of \(\phi\) every iteration.
[[Iterative Reconstruction]] – feeds errors back to modify estimator until convergence.
- steinIterativeMethodsSparse