Exact error contraction of a two-variable Gauss-Seidel iteration ($e_y^{(k+1)}=e_y^{(k)}/12$)

For the Gauss-Seidel iteration for a two-variable positive-definite system, define the forward error by $e_x^{(k)}=x^{(k)}-x_{\ast }$ and $e_y^{(k)}=y^{(k)}-y_{\ast }$. Subtract the two exact solution equations from the two update equations:The fresh first coordinate passes the old second-coordinate error through two successive factors. The iteration matrix is thereforeIts eigenvalues are $0$ and $1/12$, as seen from the determinant. The zero eigenvalue explains why the old first-coordinate error disappears completely: the first row overwrites it. After one sweep, every remaining forward error lies on the eigenvector line spanned by $(-3,1{)}^T$.

For this zero start, $e_y^{(0)}=-7/11$, so the exact vectors for $k\ge 1$ areThe Euclidean norm of the forward error decreases by exactly $1/12$ per sweep from $k=1$ onward. The first decrease is different because the initial error has not yet reached that eigenvector line. Each row update moves horizontally or vertically in the coordinate plane; sweep endpoints then approach the solution along one straight line.

The linear-system residual is an independent way to check the calculation. Completing a sweep satisfies the second row exactly, but changing $y$ disturbs the first row:Thus a zero residual in the last updated row alone does not certify a solution; inspect the complete linear-system residual.

For the same matrix, the stale schedule of the Jacobi method gives the iteration matrixHere its eigenvalues are $\pm 1/\sqrt{12}$, whereas the fresh schedule gives $\rho (H)=\rho (B{)}^2$. This comparison belongs to this particular linear system; it is not a universal relation between the two methods' spectral radii.