# Iterative Methods: Jacobi Method vs. Gauss-Seidel

In most circumstances, given a linear system: $A\mathbf{x}=\mathbf{b}$

for which G-S and Jacobi converge to the solution $\mathbf{x}$, G-S converges faster. However it is possible to concoct examples for which Jacobi method is superior. It occurs when: $$0< \rho{(G_{\text{jacobi}})} < \rho{(G_{\text{G-S}})}< 1$$ Where $G = I - Q^{-1}A$ and $Q$ is the splitting matrix defined for both methods.

Using a script in MATLAB, I came up with many integer $3\times 3$ matrices $A$ and $3\times 1$ $\mathbf{b}$ that served to be satisfying examples- although no such luck looking for a $2\times 2$ integer matrix $A$ that contributed the desired attributes to each respective $G$. I exhausted all such integer matrices $A$ with values:$-10,-9,...,9,10$. Does such an $A$ exist?? Also for our purposes $\mathbf{b}$ need not be integer valued.

-

Computing the corresponding matrices It can be seen that, in the $2\times2$ case, $\rho(G_{jacobi})^2=\rho(G_{G-S})$. This implies that the answer to your question is no.