help to find SOR optimal parameter $w$ Please let following linear system as $Ax=b$:
$$\begin{array}{l}
6a{x_1} + {x_2} + {x_3} = 1\\
{x_1} - 3a{x_2} + 4{x_3} = 2\\
{x_1} + {x_2} - 2a{x_3} = 3
\end{array}$$
Help me to prove that the optimal value $w$ in SOR method is $\dfrac{6a}{1+6a}$.
Please see https://scicomp.stackexchange.com/questions/851/are-there-any-heuristics-for-optimizing-the-successive-over-relaxation-sor-met. The eigenvalues of $D^{-1} A$ are $1-\frac{1}{3a}$, $1-\frac{1}{2a}$, and $1+\frac{5}{6a}$. Using the formula $\omega_{opt}=\frac{2}{a+b}$ in the answer, you will get the result.
The SOR method is convergent if eigenvalues are positive. Therefore a should be greater than o.5 or smaller than -5/6. The optimal parameter w=2/(1+sqrt(lambda(min)*lambda(max))). See S.M.Grzegorski, On optimal parameter not only for the SOR method, in Applied and Computational Mathematics, November 2019