Show that if the Jacobi method is convergent, then the Gauss-Seidel method is also convergent. Let A∈R2×2.
Show that if the Jacobi method applied to the system of linear equations Ax=b (with b,x∈R2 ) is convergent, then the Gauss-Seidel method is also convergent.
Does a similar relation hold in case of divergence of one of these methods? (Does divergence of one of these methods imply divergence of the other?) Justify your answer.
This is a question I came across in my class and I am not sure how to go about solving it. I know that the necessary and sufficient condition for global convergence is: ρ(B) < 1.
 A: Starting from the spectral radius of the matrix $B$ seems to be reasonable. You know that the Jacobi method converges, so the spectral radius of $B = -D^{-1}(L+U)$ is smaller than 1. Can you use this result to show that the spectral radius of $B = -(D+L)^{-1}U$ for the Gauss-Seidel method is also smaller than 1?
A: First of all, as it is mentioned in que OP, it is important to stress that the result is particular to $2\times 2$ systems. Such a result does not hold for $n>2$. Having said that, we just need to compare the spectral radius of $D^{-1}(L+U)$ and $(D+L)^{-1}U$.
$$
C_1=D^{-1}(L+U) = \begin{pmatrix} 0 & a_{12}/a_{11}\\ a_{21}/a_{22} & 0\end{pmatrix}, \quad C_2=(D+L)^{-1}U = \begin{pmatrix} 0 & a_{12}/a_{11}\\0 & -\frac{a_{12}a_{21}}{a_{11} a_{22}}\end{pmatrix}$$
So, the spectral radius of $C_1$ is
$$
\rho(C_1)=\sqrt{\left|\dfrac{a_{12} a_{21}}{a_{11}a_{22}}\right|}
$$
and the spectral radius of $C_2$ is
$$
\rho(C_2) = \left|\dfrac{a_{12} a_{21}}{a_{11}a_{22}}\right|
$$
If $\rho(C_1)<1$ the same happens to $\rho(C_2)$.
