Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 40A / iii

For $\omega =1$, the Jacobi method iteration matrix $T=I-D^{-1}A$ has zero diagonal, nonnegative axial entries $1/5$, and diagonal-neighbour entries $1/20$. Its row sums are at most one; they are strictly below one at grid points next to the Dirichlet boundary. Its graph is connected by the positive axial entries.

To prove the strict spectral radius bound, let $Tz=\lambda z$ with $z\ne 0$ and choose a component of maximum magnitude $M>0$. Thenso $|\lambda |\le 1$. If equality held, every positive-weight neighbour of a maximal component would also have modulus $M$, and that row would have sum one. Propagating along axial edges reaches a boundary-adjacent row, whose sum is strictly less than one, a contradiction. This also covers a one-point interior grid, for which $T=0$. HenceThe standard convergence theorem for a stationary linear iteration says it converges from every initial vector exactly when its iteration matrix has spectral radius below one. Since $A$ is invertible by (i), the error satisfies $e^{(r)}=T^r{e}^{(0)}\to 0$ andWeak row diagonal dominance alone would not prove convergence; the connection to a strictly deficient boundary row supplies the required strictness.