= Line-Jacobi Fourier spectrum
{title2=$\lambda_{r,s}=\cos\theta_s/(2-\cos\theta_r)$}
For the five-point Dirichlet Laplacian on an $n\times n$ square grid, a block-Jacobi splitting that solves each horizontal line exactly has blocks $\operatorname{tridiag}(-1,4,-1)$. Its tensor-product sine modes have angles $\theta_r=r\pi/(n+1)$ and the displayed <iteration matrix> eigenvalues. With $c=\cos(\pi/(n+1))$, its <spectral radius> is $c/(2-c)<1$, so it converges on every fixed grid. However the mode $r=1,s=n$ has multiplier $-c/(2-c)\to-1$. It is highly oscillatory between lines and is barely damped. The undamped method is therefore not a uniformly effective smoother for the <multigrid method>, even though it converges. Convergence and smoothing test different properties of the error spectrum.
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