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I have $A=\begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$, $b=\begin{pmatrix}3 \\ 5\end{pmatrix}$. The system to be solved is $Ax=b$. We're also given:

$$B_\theta=\frac 1 4 \begin{pmatrix} 2\theta^2 + 2\theta + 1 & -2\theta^2 + 2\theta + 1 \\ -2\theta^2 + 2\theta + 1 & 2\theta^2 + 2\theta + 1 \end{pmatrix},\quad g_\theta=\begin{pmatrix}1/2 - \theta\\ 1/2 - \theta\end{pmatrix} $$

I have to show that the iterative method

$$x_{n+1} = B_\theta x_n + g_\theta$$

converges to the solution of $Ax=b$.

I thought maybe it might be an SOR method, but I calculated the SOR matrix for $Ax=b$ and it doesn't look anything like $B_\theta$. Of course I could invert $A$ to compute $x$ and plug that into the fixed point equation, but I think we're intended to use some other method, or recognize some common iterative method.

Jack M
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  • @Variable Fixed. – Jack M Jan 09 '16 at 10:58
  • @Variable An old exercise sheet from a numerical analysis class which covered the Jacobi, Gauss-Seidel, and SOR methods. Maybe we really were just intended to see that the solution to $Ax=b$ is $(1, 1)$ and then plug that into the fixed point equation. – Jack M Jan 09 '16 at 14:20
  • Are there any more details in the problem? I implemented it and it converges for some initial points and $\theta$, but not all - so I think there is some detail missing. – Moo Jan 10 '16 at 02:43
  • @Variable The idea is to show that when $\theta$ is chosen to make it converge for all $x_0$, it converges to a solution to $Ax=b$. – Jack M Jan 10 '16 at 08:38

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Are you sure that they ask you to show that it is convergent, or only that it solves the system (consistent)? Because I just had the same problem in my university, and they did not ask to show that it always converged, but that it was consistent for any real value $θ$. Remember if an iterative method is consistent, that does not imply it is convergent.

What we had to do was to find the actual solution $x$ with any method we know (it is a simple $2*2$ linear system) and we got $x = (1,1)^t$. Then you replace it in the formula of the iterative method, both in the $x_{n+1}$ and in the $x_n$, and finally you should get the same result in both sides of the equality. That proves it is consistent.

I also got the method is convergent if $θ$ is in the interval $(-1,1/2)$, and it was considering norm of $B(θ)$ should be less than $1$.

Ronald Becerra
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