Problem set 4: Iterative solvers
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Problem set 4: Iterative solvers#
Note: If you miss the problem solving class, you have to submit solutions to all the exercises below in Studium before the deadline.
Exercise 1#
Consider the linear system of equations \(A\mathbf{x}=\mathbf{b}\), with
\[\begin{split}
A = \begin{bmatrix}
2 & -1 & 1 \\
1 & a & 2 \\
-1 & -1 & 3
\end{bmatrix},
\quad \mathbf{b} = \begin{bmatrix}
2 \\
0 \\
0
\end{bmatrix}
\end{split}\]
a) Determine all values of \(a\) for which the Jacobi method is guaranteed to converge.
b) Let \(a=3\) and \(\mathbf{x}^0 = (0,0,0)^T\). Compute \(\mathbf{x}^2\) without explicitly computing \(\mathbf{x}^1\).
c) Consider the following iterative method: Given \(\mathbf{x}^0\), set
\[
\mathbf{x}^{k+1} = M\mathbf{x}^k + \mathbf{c}, \quad k=0,1,\ldots,
\]
where
\[
M = I -\frac{1}{3}A, \quad \mathbf{c} = \frac{1}{3}\mathbf{b}.
\]
If \(a=1\), will the method converge?