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A stationary iterative method for a linear system converges from every initial vector exactly when the spectral radius of its iteration matrix is less than one. Here $\cos (kx)$ ranges between $\cos x$ and $-\cos x$, so all modes decay for a fixed $n$ exactly whenRequiring this for every grid size givesIndeed, $\omega \le 0$ leaves or amplifies the lowest mode, while any $\omega >1$ makes the highest-mode eigenvalue tend to $1-2\omega <-1$ as $n\to \infty$.

The slowest mode is $k=1$, for which the Taylor expansion isIf the initial error has a nonzero ${\mathbf{v}}_1$ component, that component is multiplied by ${\lambda }_1(\omega {)}^{\nu }$. Hence for a constant $c>0$ independent of $n$,This is the low-frequency obstruction to grid-independent convergence.

Choose a matrix splitting $A=M-N$ with $M$ invertible. Then $Ax=b$ is equivalent toand the associated stationary iterative method for a linear system isFor the Jacobi method, write $A=D+L+U$ with $D$ diagonal and $L,U$ strictly lower and upper triangular, and takeThusThe iteration converges for every initial vector exactly when the spectral radius of its iteration matrix satisfies