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For homogeneous error boundary values, the tensor-product sine vectors $U_{j,k}=\sin (r\pi jh)\sin (s\pi kh)$ form a basis. The sine addition formula gives $U_{j+1,k}+U_{j-1,k}=2\cos (r\pi h)U_{j,k}$, and similarly in the other coordinate. Hence the five-point matrix hasThere is a printing error in the displayed iteration. Taken literally, its error multiplier is $1/20-(2/5)(\cos (r\pi h)+\cos (s\pi h))$, and its fixed-point equation is $(19/4)U_{j,k}+{\sum }_{\mathrm{neighbors}}U=h^2{f}_{j,k}$, rather than the stated Poisson equation. Thus the printed affine update cannot be used as a consistent relaxation for this system.

The intended weighted Jacobi iteration with $\omega =4/5$ isIts iteration matrix is $I-(\omega /4)A$, whose mode multiplier isIf at least one of the two cosines is negative, their sum lies strictly between $-2$ and $1$, so $-3/5<{\mu }_{r,s}<3/5$. Thus every such high-frequency mode contracts by at most $\boxed{3/5}$ per sweep, uniformly in grid spacing. This weighted Jacobi smoothing bound for the square Poisson problem justifies the corrected method as a smoother. The literal printed update also damps modes, but toward the wrong fixed-point equation.

Use the fine matrix $A_h$ and right-hand side from part (b), with diagonal $D_h$. Starting from an iterate $v$, take $\nu$ weighted Jacobi smoothing stepsForm the residual $r_h=b_h-A_{h}v$. For coarse interior indices $j=1,\dots ,m/2-1$, use full-weighting restrictionThe prolongation operator matrix $P$ is linear interpolation of a coarse correction: $(Pe{)}_{2j}=e_j$ and $(Pe{)}_{2j+1}=(e_j+e_{j+1})/2$, with zero endpoint corrections. Thus $R=P^T/2$. Choose the Galerkin coarse-grid operator $A_c=R{A}_{h}P$, solve $A_c{e}_c=R{r}_h$ exactly, and correct $v\leftarrow v+P{e}_c$. The entries of $A_h$, $R$ and $P$ above specify every matrix in the cycle. A rediscretized coarse operator is another standard choice if used consistently.

For a nonsingular coarse operator, the fine-grid iteration matrix of this pre-smoothed two-grid cycle isThe purpose of under-relaxation is to damp fine-grid oscillations, leaving smoother error for coarse-grid correction. For the diffusion-dominated one-dimensional operator, $\omega =2/3$ is the usual effective choice; the high-frequency symbol $1-\omega (1-\cos \xi )$ then has modulus at most $1/3$ for $\pi /2\le |\xi |\le \pi$. Coefficients and convection can change the best parameter, so this is a motivated choice rather than a universal optimum. Under the diagonal-dominance condition, the unweighted Jacobi matrix already has norm less than one in the maximum norm, and its convex combination with the identity remains convergent for $0<\omega <1$.

The iteration matrix has eigenvalues $0,-1/2,1/2$: expanding its characteristic polynomial down the first column leaves the triangular two-by-two block with diagonal entries $-1/2,1/2$. The distinct eigenvalues ensure diagonalizability. Thus its spectral radius is $1/2<1$.

The previous bound gives $\Vert f_n\Vert \le 2^{-n}\Vert f_0\Vert \to 0$, so the Gauss-Seidel iteration converges for every initial guess under the stated diagonalization. Its limit is the exact solution $(1,1,-1{)}^T$. Since $e_n=S{f}_n$, a fixed matrix also sends this transformed error to zero; its operator norm can account for any amplification in the original coordinates.

With relaxation parameter $\omega$, weighted Jacobi updates byIts iteration matrix is $H_{\omega }=I-\omega D^{-1}A$.

Equivalently, the weighted Jacobi method takes a weighted average of the old vector and its full Jacobi method update: $x^{(k+1)}=(1-\omega )x^{(k)}+\omega x_{\mathrm{Jacobi}}^{(k+1)}$. If $Bv=\lambda v$, its iteration matrix multiplies that eigenvector by $1-\omega +\omega \lambda$. For the diagonal-4 Jacobi iteration for a three-variable tridiagonal system, the factors are $1-\omega$ and $1-\omega \pm \omega \sqrt{2}/4$; a Poisson equation smoothing parameter must not be inferred from this different matrix.