Linear-system solvers: direct vs iterative

What you are seeing: the discrete 1D Poisson problem $-u'' = \sin(\pi x)$ on $N$ grid points. The same right-hand side is fed to a Thomas direct solver (exact, $O(N)$), Jacobi, Gauss-Seidel, and conjugate gradient. The lower panel tracks the residual norm vs iteration count; CG converges in at most $N$ steps in exact arithmetic.