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 $\dfrac{6a}{1+6a}$.
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1what means the optimal value w in SOR method? – Dr. Sonnhard Graubner Jan 08 '15 at 17:17
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smallest spectral radius in SOR – SKMohammadi Jan 08 '15 at 17:25
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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.
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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
