This semester, I have been studying the most popular iterative methods, i.e., Krylov subspace iteration methods. For a large sparse system linear $$ Ax=b, $$ where $A$ is nonsingular, I know that direct methods are not suitable for this. So a number of iterative methods have been proposed, such as classic stationary iteration methods, like Jacobi, Gauss-Seidel, SOR, etc. But when the system size becomes larger, the convergence rate becomes slower and the optimal parameter for SOR is hard to choose.
Then Krylov subspace methods are required, like CG, GMRES, MINRES, etc., which are faster than those classical iteration methods. Furthermore, I have done the numerical examples in MATLAB, and the results indeed prove the faster convergence rate of Krylov methods.
My question is though Krylov subspace methods indeed are faster than classical iteration methods, but why? I mean, where can be seen that Krylov methods can be faster than those classical methods? I just know that Krylov methods must converge within $N$ iterations, for a $N$ order matrix $A$ and it does not need to choose optimal parameter. But I still do not know why Krylov methods are faster than classical iteration methods? any suggestions?