2.1 SOR 发散

定理 i-2-1（必要条件） $\bm A \bm x = \bm b$ $0 < \omega < 2$ .

证明

\begin{aligned} \det (L_{ω}) & = \det [(D - ω L)^{- 1} ((1 - ω) D + ω U)] \\ = \det [(D - ω L)^{- 1}] \det [(1 - ω) D + ω U] \\ = (1 - ω)^{n} = | λ_{1} λ_{2} \dots λ_{n} | \leq ρ (L_{ω})^{n} < 1 \\ \Leftrightarrow 0 < ω < 2. \end{aligned}

证毕.