= Semiconvergence of cyclic Poisson iterations
{title2=$A=C+C^{-1}-2I$}
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 $e^TKe/2$ 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 $w=(0,1,\ldots,1,2)^T$ has $w^T\mathbf1=n\ne0$. Neither method is a strict contraction to a unique solution without fixing the constant nullspace.
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