 Semiconvergence of cyclic Poisson iterations

The periodic second-difference circulant matrix has constant null space, so solvability requires $\sum b_j=0$. Its Jacobi method has eigenvalues $\cos (2\pi j/n)$: for odd $n$ it converges to some solution, while for even $n$ an alternating mode oscillates unless absent from the error. Lexicographic Gauss-Seidel iteration converges to some solution for every compatible right-hand side and every initial vector. For $K=-A=D+L+U$, a coordinate sweep decreases the energy $e^{T}Ke/2$ by the sum of squared coordinate changes. This excludes all nonconstant unit-modulus eigenmodes. The constant unit eigenvalue is simple, since the invariant left vector $w=(0,1,\dots ,1,2{)}^T$ has $w^{T}1=n\ne 0$. Neither method is a strict contraction to a unique solution without fixing the constant nullspace.