If 𝐮^(k) is an approximation to the solution to A 𝐮=𝐛, then the residual vector 𝐫^(k)=𝐛-A 𝐮^(k)

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If 𝐮^(k) is an approximation to the solution to A 𝐮=𝐛, then...

If $\mathbf{u}^{(k)}$ is an approximation to the solution to $A \mathbf{u}=\mathbf{b}$, then the residual vector $\mathbf{r}^{(k)}=\mathbf{b}-A \mathbf{u}^{(k)}$ measures how accurately the approximation solves the system. (a) Show that the Jacobi iteration can be written in the form $\mathbf{u}^{(k+1)}=\mathbf{u}^{(k)}+D^{-1} \mathbf{r}^{(k)}$. (b) Show that the Gauss-Seidel iteration has the form $\mathbf{u}^{(k+1)}=\mathbf{u}^{(k)}+(L+D)^{-1} \mathbf{r}^{(k)}$. (c) Show that the SOR iteration has the form $\mathbf{u}^{(k+1)}=\mathbf{u}^{(k)}+(\omega L+D)^{-1} \mathbf{r}^{(k)}$. (d) If $\| \mathbf{r}^{(k)} \mid$ is small, does this mean that $\mathbf{u}^{(k)}$ is close to the solution? Explain your answer and illustrate with a couple of examples.

If $\mathbf{u}^{(k)}$ is an approximation to the solution to $A \mathbf{u}=\mathbf{b}$, then the residual vector $\mathbf{r}^{(k)}=\mathbf{b}-A \mathbf{u}^{(k)}$ measures how accurately the approximation solves the system.
(a) Show that the Jacobi iteration can be written in the form $\mathbf{u}^{(k+1)}=\mathbf{u}^{(k)}+D^{-1} \mathbf{r}^{(k)}$.
(b) Show that the Gauss-Seidel iteration has the form $\mathbf{u}^{(k+1)}=\mathbf{u}^{(k)}+(L+D)^{-1} \mathbf{r}^{(k)}$.
(c) Show that the SOR iteration has the form $\mathbf{u}^{(k+1)}=\mathbf{u}^{(k)}+(\omega L+D)^{-1} \mathbf{r}^{(k)}$.
(d) If $\| \mathbf{r}^{(k)} \mid$ is small, does this mean that $\mathbf{u}^{(k)}$ is close to the solution? Explain your answer and illustrate with a couple of examples.

Key Concepts

Gauss-Seidel Iteration

The Gauss-Seidel iteration improves upon the Jacobi method by utilizing the lower triangular part L along with D in its update process. By using the decomposition A = L + D + U, the algorithm updates each component sequentially using already computed new values in the iteration, effectively applying u^(k+1) = u^(k) + (L+D)^(-1) r^(k), which often leads to faster convergence compared to the Jacobi method.

SOR Method

The Successive Over-Relaxation (SOR) method is an acceleration technique for iterative solvers that introduces a relaxation parameter ? to enhance convergence speed. Formulated as u^(k+1) = u^(k) + (?L+D)^(-1) r^(k), SOR generalizes the Gauss-Seidel method by scaling the influence of the lower triangular matrix term, and under appropriate conditions, the optimal choice of ? can significantly reduce the number of iterations required.

Residual Vector

The residual vector in iterative methods for solving linear systems represents the difference between the right-hand side b and the product A * u^(k) for an approximation u^(k). It quantifies the error in satisfying the system of equations and serves as a correction term in iterative methods, forming the basis for updating the current guess towards the true solution.

Jacobi Iteration

The Jacobi iteration is an iterative method for solving linear systems that uses a matrix splitting into its diagonal component D and the remainder. The update formula u^(k+1) = u^(k) + D^(-1) r^(k) uses the inverse of the diagonal matrix to correct the current approximation based on the residual, assuming that the diagonal elements are non-zero and that the method converges under suitable conditions.

Convergence Analysis in Iterative Methods

Convergence analysis in iterative solvers examines the relationship between the residual norm and the true error of the approximation. A small residual does not automatically guarantee that the approximate solution is close to the true solution, especially in cases where the system matrix is ill-conditioned. Understanding how the error propagates and the influence of the condition number of A is essential for determining the reliability of an iterative method.

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