= Alternating direction implicit iteration
{title2=$G=(\alpha I-Y)^{-1}(\alpha I+X)(\alpha I-X)^{-1}(\alpha I+Y)$}
For a rectangular <Poisson equation> discretization with commuting negative-definite directional matrices $X,Y$, two alternating implicit directional solves give the displayed iteration matrix. If their shared eigenvectors have eigenvalues $-\xi_p,-\eta_q$ with $\xi_p,\eta_q>0$, the iteration eigenvalues are $(\alpha-\xi_p)(\alpha-\eta_q)/[(\alpha+\xi_p)(\alpha+\eta_q)]$. For $\alpha>0$ their moduli are less than one, proving convergence. At $\alpha=0$ the composed iteration is the identity and the forcing terms cancel, so arbitrary initial errors persist. Strict positivity of the parameter is therefore essential for this splitting.
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