= Jacobi convergence for the nine-point Dirichlet stencil
{c}
{title2=$\rho(I-D^{-1}A)<1$}
The <Jacobi method> has axial neighbour weights $1/5$ and diagonal weights $1/20$. Interior row sums are one, boundary-adjacent sums are smaller, and the positive-weight graph is connected. If an eigenvalue had modulus one, a maximal-modulus eigenvector component would force all its neighbours to have the same modulus. Propagation reaches a deficient row, a contradiction. This proves strict spectral-radius control even though the infinity norm of the iteration matrix can equal one.
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