迭代法的收敛条件

定理 i-4-4（充要条件） 一阶定常迭代法收敛 $\iff \rho (\mathit{B})<1$$\RLA \rho(\bm B) < 1$.

证明

1. 充分性：$\rho (\mathit{B})<1\Rightarrow$$\rho(\bm B) < 1 \RA$ 迭代法收敛.

由

得

又 $\rho (\mathit{B})<1$$\rho(\bm B) < 1$，知

2. 必要性：迭代法收敛 $\Rightarrow \rho (\mathit{B})<1$$\RA \rho(\bm B) < 1$.

若 $\mathrm{\forall }{\mathit{x}}^{(0)}:\underset{k\to \mathrm{\infty }}{lim}{\mathit{x}}^{(k)}={\mathit{x}}^{\ast }$$\forall \bm x^{(0)} \colon \clim k \bm x^{(k)} = \bm x^*$，

则 $\underset{k\to \mathrm{\infty }}{lim}{\mathit{\epsilon }}^{(k)}=\underset{k\to \mathrm{\infty }}{lim}{\mathit{B}}^k{\mathit{\epsilon }}^{(0)}=\mathbf{0}$$\clim k \bm \ve^{(k)} = \clim k \bm B^{k} \bm \ve^{(0)} = \bm 0$，

于是 $\underset{k\to \mathrm{\infty }}{lim}{\mathit{B}}^k=\mathbf{0}$$\clim k \bm B^k = \bm 0$，从而 $\rho (\mathit{B})<1$$\rho(\bm B) < 1$. 证毕.