 Weighted Jacobi method for the one-dimensional Poisson equation

For the centered second-difference matrix with diagonal $-2$ and off-diagonals $1$, the weighted-Jacobi eigenvalues areThe method converges for every grid size exactly when $0<\omega \le 1$.

Multiplying both $A$ and $b$ by $-1$ leaves $D^{-1}A$ and the weighted Jacobi method unchanged. Thus the same formula applies to the positive second-difference matrix $T_n$ with diagonal $2$ and neighboring entries $-1$, the finite difference form of the one-dimensional Poisson equation with Dirichlet boundary conditions. Its eigenvectors have components $\sin (jk\pi /(n+1))$, and $D^{-1}{T}_n$ has eigenvalues $1-\cos (k\pi /(n+1))$. At $\omega =2/3$, upper-half modes have factors between $-1/3$ and $1/3$, while low-frequency factors remain close to one on fine grids. This explains why the weighted Jacobi method helps a multigrid method by reducing oscillatory errors, even when it is a slow solver on its own.^[ref]