Preparing for my Numerical Analysis exam,
If the Jacobi's method is used to solve the linear system, $Ax=b$, where
$$A=\begin{pmatrix}5 & -2 & 3\\-3 & 9 & 1\\2&-1&-7\end{pmatrix}$$
will the method be convergent?
This part I think I can do. Since there is no $b$ vector as in $Ax = b$, we cannot iterate toward a solution. Instead we reason that $A$ will converge using the Jacobi method as it is diagonally dominant.
Then it says, Determine the matrix G in the error equation $e_{n+1} = Ge_n$, and test the convergence criterion using either $||\cdot||_\infty$ or $||\cdot||_1$ norm.
I know that the norms are the max of the column and row sums, but I don't understand the "find the matrix G in the error equation" part of the question.
Best Answer
If we write $$x_{k+1}=D^{-1}(b-Rx_k)$$ where $D$ is the diagonal bit and $R$ is everything else, and we write $x_k=x+e_k$ where $e_k$ is the error in the $k$'th iterate, then $$x+e_{k+1} = D^{-1}b-D^{-1}R(x+e_k)$$ and multiplying through by $D$ we have $$Dx + De_{k+1} = b-Rx-Re_k$$ but since $x$ is a solution, $Dx+Rx=b$; consequently $$De_{k+1} = -Re_k$$ regardless of the value of $b$. Presumably you can figure out from here how to get $e_{k+1}$ by itself.